Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 13.3s

Alternatives: 30

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: 93.7% 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 93.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 2: 78.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.998:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}\\ \mathbf{elif}\;t\_4 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0.34:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_3}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (/ (sin ky) (hypot (sin ky) (sin kx)))
          (* th (fma -0.16666666666666666 (* th th) 1.0))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_2 t_3)))))
   (if (<= t_4 -0.998)
     (*
      (sin ky)
      (/
       (sin th)
       (sqrt
        (fma
         -0.5
         (cos (* ky -2.0))
         (fma (* kx kx) (fma (* kx kx) -0.3333333333333333 1.0) 0.5)))))
     (if (<= t_4 -0.05)
       t_1
       (if (<= t_4 0.34)
         (* (sin th) (/ (sin ky) (sqrt t_2)))
         (if (<= t_4 0.998)
           t_1
           (if (<= t_4 2.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_3) 1.0))
             (*
              (sin th)
              (/
               (fma ky (* -0.16666666666666666 (* ky ky)) ky)
               (hypot (sin kx) ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), sin(kx))) * (th * fma(-0.16666666666666666, (th * th), 1.0));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.998) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), fma((kx * kx), fma((kx * kx), -0.3333333333333333, 1.0), 0.5))));
	} else if (t_4 <= -0.05) {
		tmp = t_1;
	} else if (t_4 <= 0.34) {
		tmp = sin(th) * (sin(ky) / sqrt(t_2));
	} else if (t_4 <= 0.998) {
		tmp = t_1;
	} else if (t_4 <= 2.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_3), 1.0);
	} else {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / hypot(sin(kx), ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * Float64(th * fma(-0.16666666666666666, Float64(th * th), 1.0)))
	t_2 = sin(kx) ^ 2.0
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.998)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), fma(Float64(kx * kx), fma(Float64(kx * kx), -0.3333333333333333, 1.0), 0.5)))));
	elseif (t_4 <= -0.05)
		tmp = t_1;
	elseif (t_4 <= 0.34)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_2)));
	elseif (t_4 <= 0.998)
		tmp = t_1;
	elseif (t_4 <= 2.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_3), 1.0));
	else
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / hypot(sin(kx), ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = 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] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.998], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.05], t$95$1, If[LessEqual[t$95$4, 0.34], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.998], t$95$1, If[LessEqual[t$95$4, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $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))))) < -0.998

   1. Initial program 81.5%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right)}}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right) + \frac{1}{2}}}} \cdot \sin ky \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left({kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}}} \cdot \sin ky \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}}} \cdot \sin ky \]

      4. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

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

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      6. cos-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left({kx}^{2}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)}\right)}} \cdot \sin ky \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{-1}{3} \cdot {kx}^{2} + 1}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{-1}{3}} + 1, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{-1}{3}, 1\right)}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{-1}{3}, 1\right), \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      17. lower-*.f64 64.9

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.3333333333333333, 1\right), 0.5\right)\right)}} \cdot \sin ky \]

   7. Simplified 64.9%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}} \cdot \sin ky \]

   if -0.998 < (/.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))))) < -0.050000000000000003 or 0.340000000000000024 < (/.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))))) < 0.998

   1. Initial program 99.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.0

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

   7. Simplified 55.0%

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

   if -0.050000000000000003 < (/.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))))) < 0.340000000000000024

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in ky around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 93.5

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

   8. Simplified 93.5%

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

   if 0.998 < (/.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))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 99.4

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

   5. Simplified 99.4%

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

   if 2 < (/.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 2.4%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 2.4

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

   5. Simplified 2.4%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.998:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.34:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2}}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(ky \cdot -2\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.998:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, t\_1, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}\\ \mathbf{elif}\;t\_4 \leq -0.05:\\ \;\;\;\;\left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)\\ \mathbf{elif}\;t\_4 \leq 0.34:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{0.5 + \mathsf{fma}\left(-0.5, t\_1, \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_3}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* ky -2.0)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_2 t_3)))))
   (if (<= t_4 -0.998)
     (*
      (sin ky)
      (/
       (sin th)
       (sqrt
        (fma
         -0.5
         t_1
         (fma (* kx kx) (fma (* kx kx) -0.3333333333333333 1.0) 0.5)))))
     (if (<= t_4 -0.05)
       (*
        (* th (fma -0.16666666666666666 (* th th) 1.0))
        (*
         (sin ky)
         (sqrt
          (/
           1.0
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (+ 0.5 (* -0.5 (cos (+ ky ky)))))))))
       (if (<= t_4 0.34)
         (* (sin th) (/ (sin ky) (sqrt t_2)))
         (if (<= t_4 0.998)
           (*
            (sin ky)
            (*
             th
             (sqrt
              (/
               1.0
               (+ 0.5 (fma -0.5 t_1 (fma -0.5 (cos (* kx -2.0)) 0.5)))))))
           (if (<= t_4 2.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_3) 1.0))
             (*
              (sin th)
              (/
               (fma ky (* -0.16666666666666666 (* ky ky)) ky)
               (hypot (sin kx) ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((ky * -2.0));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.998) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, t_1, fma((kx * kx), fma((kx * kx), -0.3333333333333333, 1.0), 0.5))));
	} else if (t_4 <= -0.05) {
		tmp = (th * fma(-0.16666666666666666, (th * th), 1.0)) * (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))))));
	} else if (t_4 <= 0.34) {
		tmp = sin(th) * (sin(ky) / sqrt(t_2));
	} else if (t_4 <= 0.998) {
		tmp = sin(ky) * (th * sqrt((1.0 / (0.5 + fma(-0.5, t_1, fma(-0.5, cos((kx * -2.0)), 0.5))))));
	} else if (t_4 <= 2.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_3), 1.0);
	} else {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / hypot(sin(kx), ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(ky * -2.0))
	t_2 = sin(kx) ^ 2.0
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.998)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, t_1, fma(Float64(kx * kx), fma(Float64(kx * kx), -0.3333333333333333, 1.0), 0.5)))));
	elseif (t_4 <= -0.05)
		tmp = Float64(Float64(th * fma(-0.16666666666666666, Float64(th * th), 1.0)) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))))));
	elseif (t_4 <= 0.34)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_2)));
	elseif (t_4 <= 0.998)
		tmp = Float64(sin(ky) * Float64(th * sqrt(Float64(1.0 / Float64(0.5 + fma(-0.5, t_1, fma(-0.5, cos(Float64(kx * -2.0)), 0.5)))))));
	elseif (t_4 <= 2.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_3), 1.0));
	else
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / hypot(sin(kx), ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.998], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * t$95$1 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.05], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[ky], $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]), $MachinePrecision], If[LessEqual[t$95$4, 0.34], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.998], N[(N[Sin[ky], $MachinePrecision] * N[(th * N[Sqrt[N[(1.0 / N[(0.5 + N[(-0.5 * t$95$1 + N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $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))))) < -0.998

   1. Initial program 81.5%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right)}}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right) + \frac{1}{2}}}} \cdot \sin ky \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left({kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}}} \cdot \sin ky \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}}} \cdot \sin ky \]

      4. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

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

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      6. cos-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left({kx}^{2}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)}\right)}} \cdot \sin ky \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{-1}{3} \cdot {kx}^{2} + 1}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{-1}{3}} + 1, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{-1}{3}, 1\right)}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{-1}{3}, 1\right), \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      17. lower-*.f64 64.9

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.3333333333333333, 1\right), 0.5\right)\right)}} \cdot \sin ky \]

   7. Simplified 64.9%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}} \cdot \sin ky \]

   if -0.998 < (/.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))))) < -0.050000000000000003

   1. Initial program 99.3%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 50.7

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

   7. Simplified 50.7%

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

   if -0.050000000000000003 < (/.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))))) < 0.340000000000000024

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in ky around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 93.5

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

   8. Simplified 93.5%

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

   if 0.340000000000000024 < (/.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))))) < 0.998

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

   7. Simplified 56.5%

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

   if 0.998 < (/.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))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 99.4

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

   5. Simplified 99.4%

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

   if 2 < (/.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 2.4%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 2.4

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

   5. Simplified 2.4%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.998:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.34:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2}}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{0.5 + \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(ky \cdot -2\right)\\ t_2 := \mathsf{hypot}\left(\sin kx, ky\right)\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.998:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, t\_1, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}\\ \mathbf{elif}\;t\_4 \leq -0.05:\\ \;\;\;\;\left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)\\ \mathbf{elif}\;t\_4 \leq 0.2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{0.5 + \mathsf{fma}\left(-0.5, t\_1, \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_3}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* ky -2.0)))
        (t_2 (hypot (sin kx) ky))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_3)))))
   (if (<= t_4 -0.998)
     (*
      (sin ky)
      (/
       (sin th)
       (sqrt
        (fma
         -0.5
         t_1
         (fma (* kx kx) (fma (* kx kx) -0.3333333333333333 1.0) 0.5)))))
     (if (<= t_4 -0.05)
       (*
        (* th (fma -0.16666666666666666 (* th th) 1.0))
        (*
         (sin ky)
         (sqrt
          (/
           1.0
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (+ 0.5 (* -0.5 (cos (+ ky ky)))))))))
       (if (<= t_4 0.2)
         (* (sin th) (/ (sin ky) t_2))
         (if (<= t_4 0.998)
           (*
            (sin ky)
            (*
             th
             (sqrt
              (/
               1.0
               (+ 0.5 (fma -0.5 t_1 (fma -0.5 (cos (* kx -2.0)) 0.5)))))))
           (if (<= t_4 2.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_3) 1.0))
             (*
              (sin th)
              (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) t_2)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((ky * -2.0));
	double t_2 = hypot(sin(kx), ky);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_3));
	double tmp;
	if (t_4 <= -0.998) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, t_1, fma((kx * kx), fma((kx * kx), -0.3333333333333333, 1.0), 0.5))));
	} else if (t_4 <= -0.05) {
		tmp = (th * fma(-0.16666666666666666, (th * th), 1.0)) * (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))))));
	} else if (t_4 <= 0.2) {
		tmp = sin(th) * (sin(ky) / t_2);
	} else if (t_4 <= 0.998) {
		tmp = sin(ky) * (th * sqrt((1.0 / (0.5 + fma(-0.5, t_1, fma(-0.5, cos((kx * -2.0)), 0.5))))));
	} else if (t_4 <= 2.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_3), 1.0);
	} else {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / t_2);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(ky * -2.0))
	t_2 = hypot(sin(kx), ky)
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_3)))
	tmp = 0.0
	if (t_4 <= -0.998)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, t_1, fma(Float64(kx * kx), fma(Float64(kx * kx), -0.3333333333333333, 1.0), 0.5)))));
	elseif (t_4 <= -0.05)
		tmp = Float64(Float64(th * fma(-0.16666666666666666, Float64(th * th), 1.0)) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))))));
	elseif (t_4 <= 0.2)
		tmp = Float64(sin(th) * Float64(sin(ky) / t_2));
	elseif (t_4 <= 0.998)
		tmp = Float64(sin(ky) * Float64(th * sqrt(Float64(1.0 / Float64(0.5 + fma(-0.5, t_1, fma(-0.5, cos(Float64(kx * -2.0)), 0.5)))))));
	elseif (t_4 <= 2.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_3), 1.0));
	else
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / t_2));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.998], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * t$95$1 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.05], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[ky], $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]), $MachinePrecision], If[LessEqual[t$95$4, 0.2], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.998], N[(N[Sin[ky], $MachinePrecision] * N[(th * N[Sqrt[N[(1.0 / N[(0.5 + N[(-0.5 * t$95$1 + N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / t$95$2), $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))))) < -0.998

   1. Initial program 81.5%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right)}}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right) + \frac{1}{2}}}} \cdot \sin ky \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left({kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}}} \cdot \sin ky \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}}} \cdot \sin ky \]

      4. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

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

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      6. cos-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left({kx}^{2}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)}\right)}} \cdot \sin ky \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{-1}{3} \cdot {kx}^{2} + 1}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{-1}{3}} + 1, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{-1}{3}, 1\right)}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{-1}{3}, 1\right), \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      17. lower-*.f64 64.9

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.3333333333333333, 1\right), 0.5\right)\right)}} \cdot \sin ky \]

   7. Simplified 64.9%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}} \cdot \sin ky \]

   if -0.998 < (/.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))))) < -0.050000000000000003

   1. Initial program 99.3%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 50.7

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

   7. Simplified 50.7%

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

   if -0.050000000000000003 < (/.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 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 94.3

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

   5. Simplified 94.3%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 95.0

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

   7. Applied egg-rr 95.0%

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

   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))))) < 0.998

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

   7. Simplified 59.4%

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

   if 0.998 < (/.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))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 99.4

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

   5. Simplified 99.4%

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

   if 2 < (/.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 2.4%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 2.4

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

   5. Simplified 2.4%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.998:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{0.5 + \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(ky \cdot -2\right)\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ t_4 := \mathsf{hypot}\left(\sin kx, ky\right)\\ \mathbf{if}\;t\_3 \leq -0.998:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, t\_1, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;\left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)\\ \mathbf{elif}\;t\_3 \leq 0.002:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 0.998:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{0.5 + \mathsf{fma}\left(-0.5, t\_1, \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* ky -2.0)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2))))
        (t_4 (hypot (sin kx) ky)))
   (if (<= t_3 -0.998)
     (*
      (sin ky)
      (/
       (sin th)
       (sqrt
        (fma
         -0.5
         t_1
         (fma (* kx kx) (fma (* kx kx) -0.3333333333333333 1.0) 0.5)))))
     (if (<= t_3 -0.05)
       (*
        (* th (fma -0.16666666666666666 (* th th) 1.0))
        (*
         (sin ky)
         (sqrt
          (/
           1.0
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (+ 0.5 (* -0.5 (cos (+ ky ky)))))))))
       (if (<= t_3 0.002)
         (*
          (sin th)
          (/
           (*
            ky
            (fma
             (* ky ky)
             (fma
              (* ky ky)
              (fma -0.0001984126984126984 (* ky ky) 0.008333333333333333)
              -0.16666666666666666)
             1.0))
           t_4))
         (if (<= t_3 0.998)
           (*
            (sin ky)
            (*
             th
             (sqrt
              (/
               1.0
               (+ 0.5 (fma -0.5 t_1 (fma -0.5 (cos (* kx -2.0)) 0.5)))))))
           (if (<= t_3 2.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_2) 1.0))
             (*
              (sin th)
              (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) t_4)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((ky * -2.0));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double t_4 = hypot(sin(kx), ky);
	double tmp;
	if (t_3 <= -0.998) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, t_1, fma((kx * kx), fma((kx * kx), -0.3333333333333333, 1.0), 0.5))));
	} else if (t_3 <= -0.05) {
		tmp = (th * fma(-0.16666666666666666, (th * th), 1.0)) * (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))))));
	} else if (t_3 <= 0.002) {
		tmp = sin(th) * ((ky * fma((ky * ky), fma((ky * ky), fma(-0.0001984126984126984, (ky * ky), 0.008333333333333333), -0.16666666666666666), 1.0)) / t_4);
	} else if (t_3 <= 0.998) {
		tmp = sin(ky) * (th * sqrt((1.0 / (0.5 + fma(-0.5, t_1, fma(-0.5, cos((kx * -2.0)), 0.5))))));
	} else if (t_3 <= 2.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_2), 1.0);
	} else {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / t_4);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(ky * -2.0))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	t_4 = hypot(sin(kx), ky)
	tmp = 0.0
	if (t_3 <= -0.998)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, t_1, fma(Float64(kx * kx), fma(Float64(kx * kx), -0.3333333333333333, 1.0), 0.5)))));
	elseif (t_3 <= -0.05)
		tmp = Float64(Float64(th * fma(-0.16666666666666666, Float64(th * th), 1.0)) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))))));
	elseif (t_3 <= 0.002)
		tmp = Float64(sin(th) * Float64(Float64(ky * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(-0.0001984126984126984, Float64(ky * ky), 0.008333333333333333), -0.16666666666666666), 1.0)) / t_4));
	elseif (t_3 <= 0.998)
		tmp = Float64(sin(ky) * Float64(th * sqrt(Float64(1.0 / Float64(0.5 + fma(-0.5, t_1, fma(-0.5, cos(Float64(kx * -2.0)), 0.5)))))));
	elseif (t_3 <= 2.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_2), 1.0));
	else
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / t_4));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]}, If[LessEqual[t$95$3, -0.998], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * t$95$1 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.05], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[ky], $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]), $MachinePrecision], If[LessEqual[t$95$3, 0.002], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(-0.0001984126984126984 * N[(ky * ky), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.998], N[(N[Sin[ky], $MachinePrecision] * N[(th * N[Sqrt[N[(1.0 / N[(0.5 + N[(-0.5 * t$95$1 + N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / t$95$4), $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))))) < -0.998

   1. Initial program 81.5%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right)}}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right) + \frac{1}{2}}}} \cdot \sin ky \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left({kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}}} \cdot \sin ky \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}}} \cdot \sin ky \]

      4. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

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

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      6. cos-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left({kx}^{2}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)}\right)}} \cdot \sin ky \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{-1}{3} \cdot {kx}^{2} + 1}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{-1}{3}} + 1, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{-1}{3}, 1\right)}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{-1}{3}, 1\right), \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      17. lower-*.f64 64.9

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.3333333333333333, 1\right), 0.5\right)\right)}} \cdot \sin ky \]

   7. Simplified 64.9%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}} \cdot \sin ky \]

   if -0.998 < (/.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))))) < -0.050000000000000003

   1. Initial program 99.3%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 50.7

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

   7. Simplified 50.7%

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

   if -0.050000000000000003 < (/.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))))) < 2e-3

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.6

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

   5. Simplified 97.6%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 98.2

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

   7. Applied egg-rr 98.2%

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

   8. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 98.1

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

   10. Simplified 98.1%

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

   if 2e-3 < (/.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))))) < 0.998

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

   7. Simplified 57.0%

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

   if 0.998 < (/.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))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 99.4

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

   5. Simplified 99.4%

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

   if 2 < (/.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 2.4%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 2.4

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

   5. Simplified 2.4%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.998:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{0.5 + \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(ky \cdot -2\right)\\ t_2 := \cos \left(kx \cdot -2\right)\\ t_3 := \mathsf{hypot}\left(\sin kx, ky\right)\\ t_4 := {\sin ky}^{2}\\ t_5 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_4}}\\ \mathbf{if}\;t\_5 \leq -0.998:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, t\_1, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}\\ \mathbf{elif}\;t\_5 \leq -0.05:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - t\_2, \mathsf{fma}\left(-0.5, t\_1, 0.5\right)\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;t\_5 \leq 0.002:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{t\_3}\\ \mathbf{elif}\;t\_5 \leq 0.998:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{0.5 + \mathsf{fma}\left(-0.5, t\_1, \mathsf{fma}\left(-0.5, t\_2, 0.5\right)\right)}}\right)\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_4}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* ky -2.0)))
        (t_2 (cos (* kx -2.0)))
        (t_3 (hypot (sin kx) ky))
        (t_4 (pow (sin ky) 2.0))
        (t_5 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_4)))))
   (if (<= t_5 -0.998)
     (*
      (sin ky)
      (/
       (sin th)
       (sqrt
        (fma
         -0.5
         t_1
         (fma (* kx kx) (fma (* kx kx) -0.3333333333333333 1.0) 0.5)))))
     (if (<= t_5 -0.05)
       (*
        (sqrt (/ 1.0 (fma 0.5 (- 1.0 t_2) (fma -0.5 t_1 0.5))))
        (* (sin ky) th))
       (if (<= t_5 0.002)
         (*
          (sin th)
          (/
           (*
            ky
            (fma
             (* ky ky)
             (fma
              (* ky ky)
              (fma -0.0001984126984126984 (* ky ky) 0.008333333333333333)
              -0.16666666666666666)
             1.0))
           t_3))
         (if (<= t_5 0.998)
           (*
            (sin ky)
            (* th (sqrt (/ 1.0 (+ 0.5 (fma -0.5 t_1 (fma -0.5 t_2 0.5)))))))
           (if (<= t_5 2.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_4) 1.0))
             (*
              (sin th)
              (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) t_3)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((ky * -2.0));
	double t_2 = cos((kx * -2.0));
	double t_3 = hypot(sin(kx), ky);
	double t_4 = pow(sin(ky), 2.0);
	double t_5 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_4));
	double tmp;
	if (t_5 <= -0.998) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, t_1, fma((kx * kx), fma((kx * kx), -0.3333333333333333, 1.0), 0.5))));
	} else if (t_5 <= -0.05) {
		tmp = sqrt((1.0 / fma(0.5, (1.0 - t_2), fma(-0.5, t_1, 0.5)))) * (sin(ky) * th);
	} else if (t_5 <= 0.002) {
		tmp = sin(th) * ((ky * fma((ky * ky), fma((ky * ky), fma(-0.0001984126984126984, (ky * ky), 0.008333333333333333), -0.16666666666666666), 1.0)) / t_3);
	} else if (t_5 <= 0.998) {
		tmp = sin(ky) * (th * sqrt((1.0 / (0.5 + fma(-0.5, t_1, fma(-0.5, t_2, 0.5))))));
	} else if (t_5 <= 2.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_4), 1.0);
	} else {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / t_3);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(ky * -2.0))
	t_2 = cos(Float64(kx * -2.0))
	t_3 = hypot(sin(kx), ky)
	t_4 = sin(ky) ^ 2.0
	t_5 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_4)))
	tmp = 0.0
	if (t_5 <= -0.998)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, t_1, fma(Float64(kx * kx), fma(Float64(kx * kx), -0.3333333333333333, 1.0), 0.5)))));
	elseif (t_5 <= -0.05)
		tmp = Float64(sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - t_2), fma(-0.5, t_1, 0.5)))) * Float64(sin(ky) * th));
	elseif (t_5 <= 0.002)
		tmp = Float64(sin(th) * Float64(Float64(ky * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(-0.0001984126984126984, Float64(ky * ky), 0.008333333333333333), -0.16666666666666666), 1.0)) / t_3));
	elseif (t_5 <= 0.998)
		tmp = Float64(sin(ky) * Float64(th * sqrt(Float64(1.0 / Float64(0.5 + fma(-0.5, t_1, fma(-0.5, t_2, 0.5)))))));
	elseif (t_5 <= 2.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_4), 1.0));
	else
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / t_3));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -0.998], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * t$95$1 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -0.05], N[(N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - t$95$2), $MachinePrecision] + N[(-0.5 * t$95$1 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.002], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(-0.0001984126984126984 * N[(ky * ky), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.998], N[(N[Sin[ky], $MachinePrecision] * N[(th * N[Sqrt[N[(1.0 / N[(0.5 + N[(-0.5 * t$95$1 + N[(-0.5 * t$95$2 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$4), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / t$95$3), $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))))) < -0.998

   1. Initial program 81.5%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right)}}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right) + \frac{1}{2}}}} \cdot \sin ky \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left({kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}}} \cdot \sin ky \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}}} \cdot \sin ky \]

      4. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

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

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      6. cos-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + \frac{1}{2}\right)}} \cdot \sin ky \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left({kx}^{2}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)}\right)}} \cdot \sin ky \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + \frac{-1}{3} \cdot {kx}^{2}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{-1}{3} \cdot {kx}^{2} + 1}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{-1}{3}} + 1, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{-1}{3}, 1\right)}, \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{-1}{3}, 1\right), \frac{1}{2}\right)\right)}} \cdot \sin ky \]

      17. lower-*.f64 64.9

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.3333333333333333, 1\right), 0.5\right)\right)}} \cdot \sin ky \]

   7. Simplified 64.9%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}} \cdot \sin ky \]

   if -0.998 < (/.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))))) < -0.050000000000000003

   1. Initial program 99.3%

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

   4. Applied egg-rr 96.0%

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

   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. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Simplified 51.0%

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

   if -0.050000000000000003 < (/.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))))) < 2e-3

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.6

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

   5. Simplified 97.6%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 98.2

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

   7. Applied egg-rr 98.2%

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

   8. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 98.1

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

   10. Simplified 98.1%

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

   if 2e-3 < (/.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))))) < 0.998

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

   7. Simplified 57.0%

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

   if 0.998 < (/.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))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 99.4

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

   5. Simplified 99.4%

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

   if 2 < (/.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 2.4%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 2.4

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

   5. Simplified 2.4%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.998:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{0.5 + \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(ky \cdot -2\right)\\ t_2 := {\sin ky}^{2}\\ t_3 := \cos \left(kx \cdot -2\right)\\ t_4 := \mathsf{hypot}\left(\sin kx, ky\right)\\ t_5 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ t_6 := \mathsf{fma}\left(-0.5, t\_1, 0.5\right)\\ \mathbf{if}\;t\_5 \leq -1:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_6}}\\ \mathbf{elif}\;t\_5 \leq -0.05:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - t\_3, t\_6\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;t\_5 \leq 0.002:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{t\_4}\\ \mathbf{elif}\;t\_5 \leq 0.998:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{0.5 + \mathsf{fma}\left(-0.5, t\_1, \mathsf{fma}\left(-0.5, t\_3, 0.5\right)\right)}}\right)\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* ky -2.0)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (cos (* kx -2.0)))
        (t_4 (hypot (sin kx) ky))
        (t_5 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2))))
        (t_6 (fma -0.5 t_1 0.5)))
   (if (<= t_5 -1.0)
     (* (sin ky) (/ (sin th) (sqrt t_6)))
     (if (<= t_5 -0.05)
       (* (sqrt (/ 1.0 (fma 0.5 (- 1.0 t_3) t_6))) (* (sin ky) th))
       (if (<= t_5 0.002)
         (*
          (sin th)
          (/
           (*
            ky
            (fma
             (* ky ky)
             (fma
              (* ky ky)
              (fma -0.0001984126984126984 (* ky ky) 0.008333333333333333)
              -0.16666666666666666)
             1.0))
           t_4))
         (if (<= t_5 0.998)
           (*
            (sin ky)
            (* th (sqrt (/ 1.0 (+ 0.5 (fma -0.5 t_1 (fma -0.5 t_3 0.5)))))))
           (if (<= t_5 2.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_2) 1.0))
             (*
              (sin th)
              (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) t_4)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((ky * -2.0));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = cos((kx * -2.0));
	double t_4 = hypot(sin(kx), ky);
	double t_5 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double t_6 = fma(-0.5, t_1, 0.5);
	double tmp;
	if (t_5 <= -1.0) {
		tmp = sin(ky) * (sin(th) / sqrt(t_6));
	} else if (t_5 <= -0.05) {
		tmp = sqrt((1.0 / fma(0.5, (1.0 - t_3), t_6))) * (sin(ky) * th);
	} else if (t_5 <= 0.002) {
		tmp = sin(th) * ((ky * fma((ky * ky), fma((ky * ky), fma(-0.0001984126984126984, (ky * ky), 0.008333333333333333), -0.16666666666666666), 1.0)) / t_4);
	} else if (t_5 <= 0.998) {
		tmp = sin(ky) * (th * sqrt((1.0 / (0.5 + fma(-0.5, t_1, fma(-0.5, t_3, 0.5))))));
	} else if (t_5 <= 2.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_2), 1.0);
	} else {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / t_4);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(ky * -2.0))
	t_2 = sin(ky) ^ 2.0
	t_3 = cos(Float64(kx * -2.0))
	t_4 = hypot(sin(kx), ky)
	t_5 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	t_6 = fma(-0.5, t_1, 0.5)
	tmp = 0.0
	if (t_5 <= -1.0)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_6)));
	elseif (t_5 <= -0.05)
		tmp = Float64(sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - t_3), t_6))) * Float64(sin(ky) * th));
	elseif (t_5 <= 0.002)
		tmp = Float64(sin(th) * Float64(Float64(ky * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(-0.0001984126984126984, Float64(ky * ky), 0.008333333333333333), -0.16666666666666666), 1.0)) / t_4));
	elseif (t_5 <= 0.998)
		tmp = Float64(sin(ky) * Float64(th * sqrt(Float64(1.0 / Float64(0.5 + fma(-0.5, t_1, fma(-0.5, t_3, 0.5)))))));
	elseif (t_5 <= 2.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_2), 1.0));
	else
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / t_4));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(-0.5 * t$95$1 + 0.5), $MachinePrecision]}, If[LessEqual[t$95$5, -1.0], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -0.05], N[(N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - t$95$3), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.002], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(-0.0001984126984126984 * N[(ky * ky), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.998], N[(N[Sin[ky], $MachinePrecision] * N[(th * N[Sqrt[N[(1.0 / N[(0.5 + N[(-0.5 * t$95$1 + N[(-0.5 * t$95$3 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / t$95$4), $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 81.5%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 64.9

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

   7. Simplified 64.9%

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

   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))))) < -0.050000000000000003

   1. Initial program 99.3%

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

   4. Applied egg-rr 96.0%

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

   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. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Simplified 51.0%

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

   if -0.050000000000000003 < (/.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))))) < 2e-3

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.6

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

   5. Simplified 97.6%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 98.2

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

   7. Applied egg-rr 98.2%

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

   8. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 98.1

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

   10. Simplified 98.1%

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

   if 2e-3 < (/.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))))) < 0.998

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-lft-in N/A

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

   7. Simplified 57.0%

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

   if 0.998 < (/.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))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 99.4

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

   5. Simplified 99.4%

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

   if 2 < (/.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 2.4%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 2.4

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

   5. Simplified 2.4%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{0.5 + \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, ky\right)\\ t_2 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_3}}\\ t_5 := \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_2\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_4 \leq -0.05:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 0.002:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_3}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) ky))
        (t_2 (fma -0.5 (cos (* ky -2.0)) 0.5))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_3))))
        (t_5
         (*
          (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_2)))
          (* (sin ky) th))))
   (if (<= t_4 -1.0)
     (* (sin ky) (/ (sin th) (sqrt t_2)))
     (if (<= t_4 -0.05)
       t_5
       (if (<= t_4 0.002)
         (*
          (sin th)
          (/
           (*
            ky
            (fma
             (* ky ky)
             (fma
              (* ky ky)
              (fma -0.0001984126984126984 (* ky ky) 0.008333333333333333)
              -0.16666666666666666)
             1.0))
           t_1))
         (if (<= t_4 0.998)
           t_5
           (if (<= t_4 2.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_3) 1.0))
             (*
              (sin th)
              (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) t_1)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), ky);
	double t_2 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_3));
	double t_5 = sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_2))) * (sin(ky) * th);
	double tmp;
	if (t_4 <= -1.0) {
		tmp = sin(ky) * (sin(th) / sqrt(t_2));
	} else if (t_4 <= -0.05) {
		tmp = t_5;
	} else if (t_4 <= 0.002) {
		tmp = sin(th) * ((ky * fma((ky * ky), fma((ky * ky), fma(-0.0001984126984126984, (ky * ky), 0.008333333333333333), -0.16666666666666666), 1.0)) / t_1);
	} else if (t_4 <= 0.998) {
		tmp = t_5;
	} else if (t_4 <= 2.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_3), 1.0);
	} else {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / t_1);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), ky)
	t_2 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_3)))
	t_5 = Float64(sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_2))) * Float64(sin(ky) * th))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_2)));
	elseif (t_4 <= -0.05)
		tmp = t_5;
	elseif (t_4 <= 0.002)
		tmp = Float64(sin(th) * Float64(Float64(ky * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(-0.0001984126984126984, Float64(ky * ky), 0.008333333333333333), -0.16666666666666666), 1.0)) / t_1));
	elseif (t_4 <= 0.998)
		tmp = t_5;
	elseif (t_4 <= 2.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_3), 1.0));
	else
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / t_1));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = 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] * th), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.05], t$95$5, If[LessEqual[t$95$4, 0.002], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(-0.0001984126984126984 * N[(ky * ky), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.998], t$95$5, If[LessEqual[t$95$4, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / t$95$1), $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 81.5%

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

   4. Applied egg-rr 64.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 64.9

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

   7. Simplified 64.9%

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

   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))))) < -0.050000000000000003 or 2e-3 < (/.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))))) < 0.998

   1. Initial program 99.4%

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

   4. Applied egg-rr 97.6%

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

   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. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Simplified 54.0%

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

   if -0.050000000000000003 < (/.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))))) < 2e-3

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.6

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

   5. Simplified 97.6%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 98.2

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

   7. Applied egg-rr 98.2%

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

   8. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 98.1

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

   10. Simplified 98.1%

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

   if 0.998 < (/.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))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 99.4

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

   5. Simplified 99.4%

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

   if 2 < (/.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 2.4%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 2.4

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

   5. Simplified 2.4%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   10. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}} \cdot \left(\sin ky \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ t_4 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;t\_3 \leq -0.99:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{t\_4} \cdot \left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\\ \mathbf{elif}\;t\_3 \leq 0.34:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_3 \leq 0.998:\\ \;\;\;\;\sin ky \cdot \frac{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (sin ky) (* kx (fma -0.16666666666666666 (* kx kx) 1.0))))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0)))))
        (t_4 (hypot (sin ky) (sin kx))))
   (if (<= t_3 -0.99)
     t_1
     (if (<= t_3 -0.05)
       (* (/ (sin ky) t_4) (* th (fma -0.16666666666666666 (* th th) 1.0)))
       (if (<= t_3 0.34)
         (* (sin th) (/ (sin ky) (sqrt t_2)))
         (if (<= t_3 0.998)
           (* (sin ky) (/ (fma th (* -0.16666666666666666 (* th th)) th) t_4))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(sin(ky), (kx * fma(-0.16666666666666666, (kx * kx), 1.0))));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double t_4 = hypot(sin(ky), sin(kx));
	double tmp;
	if (t_3 <= -0.99) {
		tmp = t_1;
	} else if (t_3 <= -0.05) {
		tmp = (sin(ky) / t_4) * (th * fma(-0.16666666666666666, (th * th), 1.0));
	} else if (t_3 <= 0.34) {
		tmp = sin(th) * (sin(ky) / sqrt(t_2));
	} else if (t_3 <= 0.998) {
		tmp = sin(ky) * (fma(th, (-0.16666666666666666 * (th * th)), th) / t_4);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), Float64(kx * fma(-0.16666666666666666, Float64(kx * kx), 1.0)))))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	t_4 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (t_3 <= -0.99)
		tmp = t_1;
	elseif (t_3 <= -0.05)
		tmp = Float64(Float64(sin(ky) / t_4) * Float64(th * fma(-0.16666666666666666, Float64(th * th), 1.0)));
	elseif (t_3 <= 0.34)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_2)));
	elseif (t_3 <= 0.998)
		tmp = Float64(sin(ky) * Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) / t_4));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$3, -0.99], t$95$1, If[LessEqual[t$95$3, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$4), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.34], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.998], N[(N[Sin[ky], $MachinePrecision] * N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

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))))) < -0.98999999999999999 or 0.998 < (/.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 84.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.0

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

   7. Simplified 99.0%

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

   if -0.98999999999999999 < (/.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))))) < -0.050000000000000003

   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-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.7

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

   7. Simplified 55.7%

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

   if -0.050000000000000003 < (/.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))))) < 0.340000000000000024

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in ky around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 93.5

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

   8. Simplified 93.5%

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

   if 0.340000000000000024 < (/.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))))) < 0.998

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.3%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 56.2

         \[\leadsto \frac{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky \]

   9. Simplified 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.99:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.34:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2}}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\sin ky \cdot \frac{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ t_4 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\\ \mathbf{if}\;t\_3 \leq -0.99:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.34:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_3 \leq 0.998:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (sin ky) (* kx (fma -0.16666666666666666 (* kx kx) 1.0))))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0)))))
        (t_4
         (*
          (/ (sin ky) (hypot (sin ky) (sin kx)))
          (* th (fma -0.16666666666666666 (* th th) 1.0)))))
   (if (<= t_3 -0.99)
     t_1
     (if (<= t_3 -0.05)
       t_4
       (if (<= t_3 0.34)
         (* (sin th) (/ (sin ky) (sqrt t_2)))
         (if (<= t_3 0.998) t_4 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(sin(ky), (kx * fma(-0.16666666666666666, (kx * kx), 1.0))));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double t_4 = (sin(ky) / hypot(sin(ky), sin(kx))) * (th * fma(-0.16666666666666666, (th * th), 1.0));
	double tmp;
	if (t_3 <= -0.99) {
		tmp = t_1;
	} else if (t_3 <= -0.05) {
		tmp = t_4;
	} else if (t_3 <= 0.34) {
		tmp = sin(th) * (sin(ky) / sqrt(t_2));
	} else if (t_3 <= 0.998) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), Float64(kx * fma(-0.16666666666666666, Float64(kx * kx), 1.0)))))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	t_4 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * Float64(th * fma(-0.16666666666666666, Float64(th * th), 1.0)))
	tmp = 0.0
	if (t_3 <= -0.99)
		tmp = t_1;
	elseif (t_3 <= -0.05)
		tmp = t_4;
	elseif (t_3 <= 0.34)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_2)));
	elseif (t_3 <= 0.998)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.99], t$95$1, If[LessEqual[t$95$3, -0.05], t$95$4, If[LessEqual[t$95$3, 0.34], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.998], t$95$4, t$95$1]]]]]]]]

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))))) < -0.98999999999999999 or 0.998 < (/.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 84.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.0

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

   7. Simplified 99.0%

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

   if -0.98999999999999999 < (/.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))))) < -0.050000000000000003 or 0.340000000000000024 < (/.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))))) < 0.998

   1. Initial program 99.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 55.9

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

   7. Simplified 55.9%

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

   if -0.050000000000000003 < (/.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))))) < 0.340000000000000024

   1. Initial program 98.8%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.4

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

   5. Simplified 92.4%

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

   6. Taylor expanded in ky around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 93.5

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

   8. Simplified 93.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.99:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.34:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2}}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.998:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-126}:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_2 \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{t\_1}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fma ky (* -0.16666666666666666 (* ky ky)) ky))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.2)
     (*
      (* (sin ky) th)
      (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) (fma kx kx 0.5)))))
     (if (<= t_2 4e-126)
       (* t_1 (/ (sin th) (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5))))
       (if (<= t_2 1e-6)
         (* (sin th) (/ t_1 (sin kx)))
         (if (<= t_2 2.0) (sin th) (* (sin ky) (* (sin th) (/ 1.0 kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma(ky, (-0.16666666666666666 * (ky * ky)), ky);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.2) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), fma(kx, kx, 0.5))));
	} else if (t_2 <= 4e-126) {
		tmp = t_1 * (sin(th) / sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else if (t_2 <= 1e-6) {
		tmp = sin(th) * (t_1 / sin(kx));
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) * (1.0 / kx));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.2)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), fma(kx, kx, 0.5)))));
	elseif (t_2 <= 4e-126)
		tmp = Float64(t_1 * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	elseif (t_2 <= 1e-6)
		tmp = Float64(sin(th) * Float64(t_1 / sin(kx)));
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) * Float64(1.0 / kx)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $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]}, If[LessEqual[t$95$2, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(kx * kx + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-126], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(t$95$1 / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[(1.0 / kx), $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))))) < -0.20000000000000001

   1. Initial program 87.6%

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

   4. Applied egg-rr 75.4%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 45.0

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

   7. Simplified 45.0%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 23.4

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

   10. Simplified 23.4%

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

   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-126

   1. Initial program 98.7%

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

   4. Applied egg-rr 70.6%

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

   5. Taylor expanded in ky around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. cos-neg N/A

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

      16. lower-cos.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 67.6

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

   7. Simplified 67.6%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 66.5

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

   10. Simplified 66.5%

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

   if 3.9999999999999998e-126 < (/.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))))) < 9.99999999999999955e-7

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.4

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

   5. Simplified 51.4%

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

   6. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 51.4

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

   8. Simplified 51.4%

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

   if 9.99999999999999955e-7 < (/.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))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. Simplified 63.8%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.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 2.4%

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

   4. Applied egg-rr 2.4%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 2.4

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

   7. Simplified 2.4%

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

   8. Taylor expanded in kx around inf

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

      1. lower-/.f64 20.1

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

   10. Simplified 20.1%

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

       1. lift-/.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-sin.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 20.1

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

   12. Applied egg-rr 20.1%

       \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right) \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-126}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \end{array} \]

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 -0.05)
     (*
      (* (sin ky) th)
      (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) (fma kx kx 0.5)))))
     (if (<= t_1 4e-126)
       (* (* ky (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5))))
       (if (<= t_1 1e-6)
         (*
          (sin th)
          (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) (sin kx)))
         (if (<= t_1 2.0) (sin th) (* (sin ky) (* (sin th) (/ 1.0 kx)))))))))

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 <= -0.05) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), fma(kx, kx, 0.5))));
	} else if (t_1 <= 4e-126) {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else if (t_1 <= 1e-6) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) * (1.0 / kx));
	}
	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 <= -0.05)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), fma(kx, kx, 0.5)))));
	elseif (t_1 <= 4e-126)
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	elseif (t_1 <= 1e-6)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sin(kx)));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) * Float64(1.0 / kx)));
	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, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(kx * kx + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-126], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[(1.0 / kx), $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))))) < -0.050000000000000003

   1. Initial program 87.9%

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

   4. Applied egg-rr 75.9%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 44.2

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

   7. Simplified 44.2%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 22.9

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left(kx, kx, 0.5\right)}\right)}} \]

   10. Simplified 22.9%

       \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}} \]

   if -0.050000000000000003 < (/.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-126

   1. Initial program 98.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 69.8%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\cos \left(2 \cdot kx\right)\right)\right)\right)}}} \cdot \sin ky \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \left(1 + \left(\mathsf{neg}\left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)\right)\right)\right)}} \cdot \sin ky \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \left(1 + \left(\mathsf{neg}\left(\cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}\right)\right)\right)}} \cdot \sin ky \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)}}} \cdot \sin ky \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2}} + \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)}} \cdot \sin ky \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)}}} \cdot \sin ky \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \sin ky \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \sin ky \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot kx\right)}}} \cdot \sin ky \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\color{blue}{2} \cdot kx\right)}} \cdot \sin ky \]

      11. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2}}}} \cdot \sin ky \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \cdot \sin ky \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \cdot \sin ky \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \cdot \sin ky \]

      15. cos-neg N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      17. *-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      18. lower-*.f64 68.7

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \cdot \sin ky \]

   7. Simplified 68.7%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \cdot \sin ky \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \sin th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \left(ky \cdot \color{blue}{\sin th}\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}}} \]

      6. +-commutative N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right) + \frac{1}{2}}}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(-2 \cdot kx\right), \frac{1}{2}\right)}}} \]

      8. lower-cos.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      9. *-commutative N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      10. lower-*.f64 68.2

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   10. Simplified 68.2%

       \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]

   if 3.9999999999999998e-126 < (/.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))))) < 9.99999999999999955e-7

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.4

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 51.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sin kx} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}{\sin kx} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}}{\sin kx} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}}{\sin kx} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}}{\sin kx} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\sin kx} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\sin kx} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right)}{\sin kx} \cdot \sin th \]

      8. lower-*.f64 51.4

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right)}{\sin kx} \cdot \sin th \]

   8. Simplified 51.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}}{\sin kx} \cdot \sin th \]

   if 9.99999999999999955e-7 < (/.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))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. Simplified 63.8%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.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 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 2.4

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 2.4%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in kx around inf

      \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 20.1

         \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

   10. Simplified 20.1%

       \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

       2. lift-sin.f64 N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{kx}\right)} \cdot \sin th \]

       4. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot \frac{1}{kx}\right) \cdot \color{blue}{\sin th} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]

       7. lower-*.f64 20.1

          \[\leadsto \sin ky \cdot \color{blue}{\left(\frac{1}{kx} \cdot \sin th\right)} \]

   12. Applied egg-rr 20.1%

       \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-126}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.15:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{\sin kx}\right)\\ \end{array} \end{array} \]

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 -0.98)
     (*
      (* (sin ky) th)
      (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) (fma kx kx 0.5)))))
     (if (<= t_1 0.15)
       (* (sin ky) (/ (sin th) (sin kx)))
       (if (<= t_1 2.0)
         (sin th)
         (* (sin ky) (* (sin th) (/ 1.0 (sin kx)))))))))

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 <= -0.98) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), fma(kx, kx, 0.5))));
	} else if (t_1 <= 0.15) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) * (1.0 / sin(kx)));
	}
	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 <= -0.98)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), fma(kx, kx, 0.5)))));
	elseif (t_1 <= 0.15)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) * Float64(1.0 / sin(kx))));
	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, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(kx * kx + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.15], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))) < -0.97999999999999998

   1. Initial program 82.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 65.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 62.9

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 62.9%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]

      7. +-commutative N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right) + \frac{1}{2}}}} \]

      8. associate-+l+ N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left({kx}^{2} + \frac{1}{2}\right)}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), {kx}^{2} + \frac{1}{2}\right)}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), {kx}^{2} + \frac{1}{2}\right)}} \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      12. cos-neg N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      14. *-commutative N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{kx \cdot kx} + \frac{1}{2}\right)}} \]

      17. lower-fma.f64 31.6

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left(kx, kx, 0.5\right)}\right)}} \]

   10. Simplified 31.6%

       \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}} \]

   if -0.97999999999999998 < (/.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))))) < 0.149999999999999994

   1. Initial program 98.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 42.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 42.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky}}{\sin kx} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \color{blue}{\sin th} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      7. lower-/.f64 42.1

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   7. Applied egg-rr 42.1%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

   if 0.149999999999999994 < (/.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))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 66.5%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.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 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 20.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 20.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky}}{\sin kx} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\sin kx}\right)} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot \frac{1}{\sin kx}\right) \cdot \color{blue}{\sin th} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{\sin kx} \cdot \sin th\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{\sin kx} \cdot \sin th\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sin ky \cdot \color{blue}{\left(\frac{1}{\sin kx} \cdot \sin th\right)} \]

      8. lower-/.f64 20.1

         \[\leadsto \sin ky \cdot \left(\color{blue}{\frac{1}{\sin kx}} \cdot \sin th\right) \]

   7. Applied egg-rr 20.1%

      \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{\sin kx} \cdot \sin th\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.15:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{\sin kx}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.15:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \end{array} \]

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 -0.98)
     (*
      (* (sin ky) th)
      (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) (fma kx kx 0.5)))))
     (if (<= t_1 0.15)
       (* (sin ky) (/ (sin th) (sin kx)))
       (if (<= t_1 2.0) (sin th) (* (sin ky) (* (sin th) (/ 1.0 kx))))))))

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 <= -0.98) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), fma(kx, kx, 0.5))));
	} else if (t_1 <= 0.15) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) * (1.0 / kx));
	}
	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 <= -0.98)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), fma(kx, kx, 0.5)))));
	elseif (t_1 <= 0.15)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) * Float64(1.0 / kx)));
	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, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(kx * kx + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.15], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[(1.0 / kx), $MachinePrecision]), $MachinePrecision]), $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))))) < -0.97999999999999998

   1. Initial program 82.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 65.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 62.9

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 62.9%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]

      7. +-commutative N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right) + \frac{1}{2}}}} \]

      8. associate-+l+ N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left({kx}^{2} + \frac{1}{2}\right)}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), {kx}^{2} + \frac{1}{2}\right)}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), {kx}^{2} + \frac{1}{2}\right)}} \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      12. cos-neg N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      14. *-commutative N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{kx \cdot kx} + \frac{1}{2}\right)}} \]

      17. lower-fma.f64 31.6

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left(kx, kx, 0.5\right)}\right)}} \]

   10. Simplified 31.6%

       \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}} \]

   if -0.97999999999999998 < (/.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))))) < 0.149999999999999994

   1. Initial program 98.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 42.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 42.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky}}{\sin kx} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \color{blue}{\sin th} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      7. lower-/.f64 42.1

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   7. Applied egg-rr 42.1%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

   if 0.149999999999999994 < (/.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))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 66.5%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.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 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 2.4

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 2.4%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in kx around inf

      \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 20.1

         \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

   10. Simplified 20.1%

       \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

       2. lift-sin.f64 N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{kx}\right)} \cdot \sin th \]

       4. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot \frac{1}{kx}\right) \cdot \color{blue}{\sin th} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]

       7. lower-*.f64 20.1

          \[\leadsto \sin ky \cdot \color{blue}{\left(\frac{1}{kx} \cdot \sin th\right)} \]

   12. Applied egg-rr 20.1%

       \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.15:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-126}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \end{array} \]

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 4e-126)
     (* (* ky (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5))))
     (if (<= t_1 1e-6)
       (* (sin th) (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) (sin kx)))
       (if (<= t_1 2.0) (sin th) (* (sin ky) (* (sin th) (/ 1.0 kx))))))))

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 <= 4e-126) {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else if (t_1 <= 1e-6) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) * (1.0 / kx));
	}
	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 <= 4e-126)
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	elseif (t_1 <= 1e-6)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sin(kx)));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) * Float64(1.0 / kx)));
	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, 4e-126], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[(1.0 / kx), $MachinePrecision]), $MachinePrecision]), $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))))) < 3.9999999999999998e-126

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\cos \left(2 \cdot kx\right)\right)\right)\right)}}} \cdot \sin ky \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \left(1 + \left(\mathsf{neg}\left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)\right)\right)\right)}} \cdot \sin ky \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \left(1 + \left(\mathsf{neg}\left(\cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}\right)\right)\right)}} \cdot \sin ky \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)}}} \cdot \sin ky \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2}} + \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)}} \cdot \sin ky \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)}}} \cdot \sin ky \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \sin ky \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \sin ky \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot kx\right)}}} \cdot \sin ky \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\color{blue}{2} \cdot kx\right)}} \cdot \sin ky \]

      11. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2}}}} \cdot \sin ky \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \cdot \sin ky \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \cdot \sin ky \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \cdot \sin ky \]

      15. cos-neg N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      17. *-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      18. lower-*.f64 35.6

          \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \cdot \sin ky \]

   7. Simplified 35.6%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \cdot \sin ky \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \sin th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \left(ky \cdot \color{blue}{\sin th}\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right)}}} \]

      6. +-commutative N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{-1}{2} \cdot \cos \left(-2 \cdot kx\right) + \frac{1}{2}}}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(-2 \cdot kx\right), \frac{1}{2}\right)}}} \]

      8. lower-cos.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      9. *-commutative N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      10. lower-*.f64 31.8

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   10. Simplified 31.8%

       \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]

   if 3.9999999999999998e-126 < (/.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))))) < 9.99999999999999955e-7

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.4

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 51.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sin kx} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}{\sin kx} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}}{\sin kx} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}}{\sin kx} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}}{\sin kx} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\sin kx} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\sin kx} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right)}{\sin kx} \cdot \sin th \]

      8. lower-*.f64 51.4

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right)}{\sin kx} \cdot \sin th \]

   8. Simplified 51.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}}{\sin kx} \cdot \sin th \]

   if 9.99999999999999955e-7 < (/.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))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. Simplified 63.8%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.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 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 2.4

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 2.4%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in kx around inf

      \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 20.1

         \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

   10. Simplified 20.1%

       \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

       2. lift-sin.f64 N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{kx}\right)} \cdot \sin th \]

       4. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot \frac{1}{kx}\right) \cdot \color{blue}{\sin th} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]

       7. lower-*.f64 20.1

          \[\leadsto \sin ky \cdot \color{blue}{\left(\frac{1}{kx} \cdot \sin th\right)} \]

   12. Applied egg-rr 20.1%

       \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-126}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \end{array} \]

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 1e-6)
     (* (sin th) (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) (sin kx)))
     (if (<= t_1 2.0) (sin th) (* (sin ky) (* (sin th) (/ 1.0 kx)))))))

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 <= 1e-6) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) * (1.0 / kx));
	}
	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 <= 1e-6)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sin(kx)));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) * Float64(1.0 / kx)));
	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, 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[(1.0 / kx), $MachinePrecision]), $MachinePrecision]), $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))))) < 9.99999999999999955e-7

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 32.0

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 32.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sin kx} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}{\sin kx} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}}{\sin kx} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}}{\sin kx} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}}{\sin kx} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\sin kx} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\sin kx} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right)}{\sin kx} \cdot \sin th \]

      8. lower-*.f64 30.3

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right)}{\sin kx} \cdot \sin th \]

   8. Simplified 30.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}}{\sin kx} \cdot \sin th \]

   if 9.99999999999999955e-7 < (/.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))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. Simplified 63.8%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.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 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 2.4

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 2.4%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in kx around inf

      \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 20.1

         \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

   10. Simplified 20.1%

       \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

       2. lift-sin.f64 N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{kx}\right)} \cdot \sin th \]

       4. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot \frac{1}{kx}\right) \cdot \color{blue}{\sin th} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]

       7. lower-*.f64 20.1

          \[\leadsto \sin ky \cdot \color{blue}{\left(\frac{1}{kx} \cdot \sin th\right)} \]

   12. Applied egg-rr 20.1%

       \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \end{array} \]

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 1e-6)
     (* (sin th) (/ ky (sin kx)))
     (if (<= t_1 2.0) (sin th) (* (sin ky) (* (sin th) (/ 1.0 kx)))))))

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 <= 1e-6) {
		tmp = sin(th) * (ky / sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(ky) * (sin(th) * (1.0 / kx));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= 1d-6) then
        tmp = sin(th) * (ky / sin(kx))
    else if (t_1 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = sin(ky) * (sin(th) * (1.0d0 / kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 1e-6) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (Math.sin(th) * (1.0 / kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 1e-6:
		tmp = math.sin(th) * (ky / math.sin(kx))
	elif t_1 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = math.sin(ky) * (math.sin(th) * (1.0 / kx))
	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 <= 1e-6)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(ky) * Float64(sin(th) * Float64(1.0 / kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 1e-6)
		tmp = sin(th) * (ky / sin(kx));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = sin(ky) * (sin(th) * (1.0 / kx));
	end
	tmp_2 = 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, 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[(1.0 / kx), $MachinePrecision]), $MachinePrecision]), $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))))) < 9.99999999999999955e-7

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 30.6

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 9.99999999999999955e-7 < (/.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))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. Simplified 63.8%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.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 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 2.4

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 2.4%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in kx around inf

      \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 20.1

         \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

   10. Simplified 20.1%

       \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

       2. lift-sin.f64 N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{kx}\right)} \cdot \sin th \]

       4. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot \frac{1}{kx}\right) \cdot \color{blue}{\sin th} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]

       7. lower-*.f64 20.1

          \[\leadsto \sin ky \cdot \color{blue}{\left(\frac{1}{kx} \cdot \sin th\right)} \]

   12. Applied egg-rr 20.1%

       \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{kx} \cdot \sin th\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left(\sin th \cdot \frac{1}{kx}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \end{array} \end{array} \]

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 1e-6)
     (* (sin th) (/ ky (sin kx)))
     (if (<= t_1 2.0) (sin th) (* (sin th) (/ ky kx))))))

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 <= 1e-6) {
		tmp = sin(th) * (ky / sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (ky / kx);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= 1d-6) then
        tmp = sin(th) * (ky / sin(kx))
    else if (t_1 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = sin(th) * (ky / kx)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 1e-6) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / kx);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 1e-6:
		tmp = math.sin(th) * (ky / math.sin(kx))
	elif t_1 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = math.sin(th) * (ky / kx)
	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 <= 1e-6)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(ky / kx));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 1e-6)
		tmp = sin(th) * (ky / sin(kx));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = sin(th) * (ky / kx);
	end
	tmp_2 = 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, 1e-6], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $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))))) < 9.99999999999999955e-7

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 30.6

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 9.99999999999999955e-7 < (/.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))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. Simplified 63.8%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.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 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 2.4

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 2.4%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 20.1

         \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]

   10. Simplified 20.1%

       \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{0.5 + \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2e-9)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (* kx (fma -0.16666666666666666 (* kx kx) 1.0)))))
   (*
    (sin ky)
    (/
     (sin th)
     (sqrt
      (+
       0.5
       (fma -0.5 (cos (* ky -2.0)) (fma -0.5 (cos (* kx -2.0)) 0.5))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 2e-9) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), (kx * fma(-0.16666666666666666, (kx * kx), 1.0))));
	} else {
		tmp = sin(ky) * (sin(th) / sqrt((0.5 + fma(-0.5, cos((ky * -2.0)), fma(-0.5, cos((kx * -2.0)), 0.5)))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 2e-9)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), Float64(kx * fma(-0.16666666666666666, Float64(kx * kx), 1.0)))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(Float64(0.5 + fma(-0.5, cos(Float64(ky * -2.0)), fma(-0.5, cos(Float64(kx * -2.0)), 0.5))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 2e-9], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(0.5 + N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.00000000000000012e-9

   1. Initial program 86.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      5. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 99.9

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)}\right)} \cdot \sin th \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)}\right)} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right)\right)} \cdot \sin th \]

      5. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right)\right)} \cdot \sin th \]

   7. Simplified 99.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)}\right)} \cdot \sin th \]

   if 2.00000000000000012e-9 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky} \]

   5. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin ky \]

      2. sub-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\cos \left(2 \cdot kx\right)\right)\right)\right)}\right)}} \cdot \sin ky \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 + \left(\mathsf{neg}\left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)\right)\right)\right)\right)}} \cdot \sin ky \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \left(1 + \left(\mathsf{neg}\left(\cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}\right)\right)\right)\right)}} \cdot \sin ky \]

      5. sub-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)}\right)}} \cdot \sin ky \]

      6. sub-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)\right)}\right)}} \cdot \sin ky \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \color{blue}{\left(\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)\right)}\right)}} \cdot \sin ky \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left(\color{blue}{\frac{1}{2}} + \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)\right)\right)}} \cdot \sin ky \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)}\right)\right)}} \cdot \sin ky \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}\right)\right)}} \cdot \sin ky \]

      11. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left(\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)\right)}} \cdot \sin ky \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot kx\right)}\right)\right)}} \cdot \sin ky \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\color{blue}{2} \cdot kx\right)\right)\right)}} \cdot \sin ky \]

   7. Simplified 99.1%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{0.5 + \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}} \cdot \sin ky \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{0.5 + \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2e-9)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (* kx (fma -0.16666666666666666 (* kx kx) 1.0)))))
   (*
    (sin th)
    (/
     (sin ky)
     (sqrt
      (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 (pow(sin(kx), 2.0) <= 2e-9) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), (kx * fma(-0.16666666666666666, (kx * kx), 1.0))));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt(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 ((sin(kx) ^ 2.0) <= 2e-9)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), Float64(kx * fma(-0.16666666666666666, Float64(kx * kx), 1.0)))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(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[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 2e-9], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.00000000000000012e-9

   1. Initial program 86.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      5. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 99.9

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)}\right)} \cdot \sin th \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)}\right)} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right)\right)} \cdot \sin th \]

      5. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right)\right)} \cdot \sin th \]

   7. Simplified 99.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)}\right)} \cdot \sin th \]

   if 2.00000000000000012e-9 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sqrt.f64 99.4

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      12. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      13. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      14. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}}} \cdot \sin th \]

   4. Applied egg-rr 99.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 15.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-308}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      2e-308)
   (* -0.16666666666666666 (* th (* 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))))) <= 2e-308) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = 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(th) * (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))))) <= 2d-308) then
        tmp = (-0.16666666666666666d0) * (th * (th * th))
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(th) * (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))))) <= 2e-308) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(th) * (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))))) <= 2e-308:
		tmp = -0.16666666666666666 * (th * (th * th))
	else:
		tmp = 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))))) <= 2e-308)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(th) * (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 2e-308)
		tmp = -0.16666666666666666 * (th * (th * th));
	else
		tmp = th;
	end
	tmp_2 = 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], 2e-308], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], th]

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)) < 1.9999999999999998e-308

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 23.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 23.3%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

      5. lower-*.f64 11.7

         \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

   8. Simplified 11.7%

      \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot \left(th \cdot th\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{{th}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot {th}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       6. lower-*.f64 11.3

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 11.3%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 1.9999999999999998e-308 < (*.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 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 21.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 21.4%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

      5. lower-*.f64 14.4

         \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

   8. Simplified 14.4%

      \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

   9. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 14.7%

       \[\leadsto th \cdot \color{blue}{1} \]
   12. Step-by-step derivation

       1. *-rgt-identity 14.7

          \[\leadsto \color{blue}{th} \]

   13. Applied egg-rr 14.7%

       \[\leadsto \color{blue}{th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-308}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 30.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 3 \cdot 10^{-45}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 3e-45)
   (* -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)))) <= 3e-45) {
		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)))) <= 3d-45) 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)))) <= 3e-45) {
		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)))) <= 3e-45:
		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)))) <= 3e-45)
		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)))) <= 3e-45)
		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], 3e-45], 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))))) < 3.00000000000000011e-45

   1. Initial program 92.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 3.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

      5. lower-*.f64 3.4

         \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

   8. Simplified 3.4%

      \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot \left(th \cdot th\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{{th}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot {th}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       6. lower-*.f64 10.0

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 10.0%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 3.00000000000000011e-45 < (/.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 93.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 58.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 70.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;\sin ky \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.1)
   (*
    (* (sin ky) th)
    (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) (fma kx kx 0.5)))))
   (if (<= (sin ky) 0.001)
     (*
      (sin th)
      (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) (hypot (sin kx) ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.1) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), fma(kx, kx, 0.5))));
	} else if (sin(ky) <= 0.001) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / hypot(sin(kx), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.1)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), fma(kx, kx, 0.5)))));
	elseif (sin(ky) <= 0.001)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / hypot(sin(kx), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.1], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(kx * kx + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.001], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.10000000000000001

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 57.7

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 57.7%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]

      7. +-commutative N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right) + \frac{1}{2}}}} \]

      8. associate-+l+ N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left({kx}^{2} + \frac{1}{2}\right)}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), {kx}^{2} + \frac{1}{2}\right)}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), {kx}^{2} + \frac{1}{2}\right)}} \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      12. cos-neg N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      14. *-commutative N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} + \frac{1}{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{kx \cdot kx} + \frac{1}{2}\right)}} \]

      17. lower-fma.f64 29.8

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left(kx, kx, 0.5\right)}\right)}} \]

   10. Simplified 29.8%

       \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}} \]

   if -0.10000000000000001 < (sin.f64 ky) < 1e-3

   1. Initial program 85.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

      2. lower-*.f64 83.9

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   5. Simplified 83.9%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + ky \cdot ky}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + ky \cdot ky}} \cdot \sin th \]

      3. lower-hypot.f64 97.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, ky\right)}} \cdot \sin th \]

   7. Applied egg-rr 97.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, ky\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\frac{-1}{6} \cdot {ky}^{2}}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      7. lower-*.f64 97.5

         \[\leadsto \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   10. Simplified 97.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   if 1e-3 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 58.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;\sin ky \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 64.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0066:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0066)
   (*
    (sin th)
    (/
     (*
      ky
      (fma
       (* ky ky)
       (fma
        (* ky ky)
        (fma -0.0001984126984126984 (* ky ky) 0.008333333333333333)
        -0.16666666666666666)
       1.0))
     (hypot (sin kx) ky)))
   (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0066) {
		tmp = sin(th) * ((ky * fma((ky * ky), fma((ky * ky), fma(-0.0001984126984126984, (ky * ky), 0.008333333333333333), -0.16666666666666666), 1.0)) / hypot(sin(kx), ky));
	} else {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0066)
		tmp = Float64(sin(th) * Float64(Float64(ky * fma(Float64(ky * ky), fma(Float64(ky * ky), fma(-0.0001984126984126984, Float64(ky * ky), 0.008333333333333333), -0.16666666666666666), 1.0)) / hypot(sin(kx), ky)));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0066], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(-0.0001984126984126984 * N[(ky * ky), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0066

   1. Initial program 90.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

      2. lower-*.f64 57.3

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   5. Simplified 57.3%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + ky \cdot ky}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + ky \cdot ky}} \cdot \sin th \]

      3. lower-hypot.f64 66.4

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, ky\right)}} \cdot \sin th \]

   7. Applied egg-rr 66.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, ky\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right) + 1\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, {ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}, 1\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}, 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, {ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}, 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      6. sub-neg N/A

         \[\leadsto \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{{ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, {ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right)}, 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      11. +-commutative N/A

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-1}{5040} \cdot {ky}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {ky}^{2}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{ky \cdot ky}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      14. lower-*.f64 64.8

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, \color{blue}{ky \cdot ky}, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   10. Simplified 64.8%

       \[\leadsto \frac{\color{blue}{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   if 0.0066 < ky

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \cdot \sin ky \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), \frac{1}{2}\right)}}} \cdot \sin ky \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), \frac{1}{2}\right)}} \cdot \sin ky \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      8. lower-*.f64 60.5

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \cdot \sin ky \]

   7. Simplified 60.5%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \cdot \sin ky \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0066:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 64.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0066:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0066)
   (*
    (sin th)
    (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) (hypot (sin kx) ky)))
   (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0066) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / hypot(sin(kx), ky));
	} else {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0066)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / hypot(sin(kx), ky)));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0066], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0066

   1. Initial program 90.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

      2. lower-*.f64 57.3

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   5. Simplified 57.3%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + ky \cdot ky}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + ky \cdot ky}} \cdot \sin th \]

      3. lower-hypot.f64 66.4

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, ky\right)}} \cdot \sin th \]

   7. Applied egg-rr 66.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, ky\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\frac{-1}{6} \cdot {ky}^{2}}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

      7. lower-*.f64 64.9

         \[\leadsto \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   10. Simplified 64.9%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin th \]

   if 0.0066 < ky

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \cdot \sin ky \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), \frac{1}{2}\right)}}} \cdot \sin ky \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), \frac{1}{2}\right)}} \cdot \sin ky \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \cdot \sin ky \]

      8. lower-*.f64 60.5

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \cdot \sin ky \]

   7. Simplified 60.5%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \cdot \sin ky \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0066:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 34.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1.45e-114) (* (sin ky) (/ (sin th) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1.45e-114) {
		tmp = sin(ky) * (sin(th) / 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) <= 1.45d-114) then
        tmp = sin(ky) * (sin(th) / 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) <= 1.45e-114) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 1.45e-114:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1.45e-114)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 1.45e-114)
		tmp = sin(ky) * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1.45e-114], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 1.44999999999999998e-114

   1. Initial program 89.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 30.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 30.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{kx}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{kx}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{kx} \]

      4. lower-/.f64 N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{kx}} \]

      5. lower-sin.f64 20.6

         \[\leadsto \sin ky \cdot \frac{\color{blue}{\sin th}}{kx} \]

   8. Simplified 20.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{kx}} \]

   if 1.44999999999999998e-114 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 57.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 57.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 33.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;\sin th \cdot \left(\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right) \cdot \frac{1}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 1.45e-114)
   (* (sin th) (* (fma ky (* -0.16666666666666666 (* ky ky)) ky) (/ 1.0 kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 1.45e-114) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) * (1.0 / kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 1.45e-114)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) * Float64(1.0 / kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 1.45e-114], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] * N[(1.0 / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 1.44999999999999998e-114

   1. Initial program 89.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 69.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 44.0

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 44.0%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in kx around inf

      \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 20.6

         \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

   10. Simplified 20.6%

       \[\leadsto \left(\color{blue}{\frac{1}{kx}} \cdot \sin ky\right) \cdot \sin th \]

   11. Taylor expanded in ky around 0

       \[\leadsto \left(\frac{1}{kx} \cdot \color{blue}{\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)\right)}\right) \cdot \sin th \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}\right)\right) \cdot \sin th \]

       2. distribute-lft-in N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \color{blue}{\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1\right)}\right) \cdot \sin th \]

       3. *-rgt-identity N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}\right)\right) \cdot \sin th \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}\right) \cdot \sin th \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \mathsf{fma}\left(ky, \color{blue}{\frac{-1}{6} \cdot {ky}^{2}}, ky\right)\right) \cdot \sin th \]

       6. unpow2 N/A

          \[\leadsto \left(\frac{1}{kx} \cdot \mathsf{fma}\left(ky, \frac{-1}{6} \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)\right) \cdot \sin th \]

       7. lower-*.f64 19.7

          \[\leadsto \left(\frac{1}{kx} \cdot \mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)\right) \cdot \sin th \]

   13. Simplified 19.7%

       \[\leadsto \left(\frac{1}{kx} \cdot \color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}\right) \cdot \sin th \]

   if 1.44999999999999998e-114 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 57.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 57.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;\sin th \cdot \left(\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right) \cdot \frac{1}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 26.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.45e-114) (* (sin th) (/ ky kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.45e-114) {
		tmp = sin(th) * (ky / 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 (ky <= 1.45d-114) then
        tmp = sin(th) * (ky / 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 (ky <= 1.45e-114) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.45e-114:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.45e-114)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.45e-114)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.45e-114], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.44999999999999998e-114

   1. Initial program 89.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 43.2

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 43.2%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 20.5

         \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]

   10. Simplified 20.5%

       \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]

   if 1.44999999999999998e-114 < ky

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 36.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 36.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 26.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.45e-114) (* ky (/ (sin th) kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.45e-114) {
		tmp = ky * (sin(th) / 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 (ky <= 1.45d-114) then
        tmp = ky * (sin(th) / 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 (ky <= 1.45e-114) {
		tmp = ky * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 1.45e-114:
		tmp = ky * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.45e-114)
		tmp = Float64(ky * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 1.45e-114)
		tmp = ky * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.45e-114], N[(ky * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.44999999999999998e-114

   1. Initial program 89.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot {kx}^{2}}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 43.2

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \color{blue}{\left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 43.2%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{2 \cdot \left(kx \cdot kx\right)}, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{kx}} \]

      3. lower-/.f64 N/A

         \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{kx}} \]

      4. lower-sin.f64 20.6

         \[\leadsto ky \cdot \frac{\color{blue}{\sin th}}{kx} \]

   10. Simplified 20.6%

       \[\leadsto \color{blue}{ky \cdot \frac{\sin th}{kx}} \]

   if 1.44999999999999998e-114 < ky

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 36.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 36.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 13.4% accurate, 632.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return 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 = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 22.4

      \[\leadsto \color{blue}{\sin th} \]

5. Simplified 22.4%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

   4. unpow2 N/A

      \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

   5. lower-*.f64 13.0

      \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

8. Simplified 13.0%

   \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

9. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{1} \]
10. Step-by-step derivation

11. Simplified 13.4%

    \[\leadsto th \cdot \color{blue}{1} \]
12. Step-by-step derivation

    1. *-rgt-identity 13.4

       \[\leadsto \color{blue}{th} \]

13. Applied egg-rr 13.4%

    \[\leadsto \color{blue}{th} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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)))