Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.7%

Time: 17.8s

Alternatives: 34

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% 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 93.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 2: 87.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, kx \cdot kx, kx\right)\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{t\_3} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 0.9981109276908842:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ (sin ky) (hypot (sin ky) (fma kx (* kx kx) kx)))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (hypot (sin ky) (sin kx))))
   (if (<= t_2 -1.0)
     t_1
     (if (<= t_2 -0.01)
       (* (/ (sin ky) t_3) (fma th (* -0.16666666666666666 (* th th)) th))
       (if (<= t_2 2e-7)
         (* (sin th) (/ (fma ky (* -0.16666666666666666 (* ky ky)) ky) t_3))
         (if (<= t_2 0.9981109276908842)
           (*
            (* (sin ky) th)
            (sqrt
             (/
              1.0
              (fma
               0.5
               (- 1.0 (cos (* ky -2.0)))
               (fma -0.5 (cos (* kx -2.0)) 0.5)))))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (kx * kx), kx)));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = hypot(sin(ky), sin(kx));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.01) {
		tmp = (sin(ky) / t_3) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else if (t_2 <= 2e-7) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / t_3);
	} else if (t_2 <= 0.9981109276908842) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(0.5, (1.0 - cos((ky * -2.0))), fma(-0.5, cos((kx * -2.0)), 0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(kx * kx), kx))))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.01)
		tmp = Float64(Float64(sin(ky) / t_3) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	elseif (t_2 <= 2e-7)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / t_3));
	elseif (t_2 <= 0.9981109276908842)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(ky * -2.0))), fma(-0.5, cos(Float64(kx * -2.0)), 0.5)))));
	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[(kx * kx), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], t$95$1, If[LessEqual[t$95$2, -0.01], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$3), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-7], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9981109276908842], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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))))) < -1 or 0.99811092769088416 < (/.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 84.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 100.0

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

   4. Applied egg-rr 100.0%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   10. Simplified 100.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0100000000000000002

   1. Initial program 99.5%

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

      1. lift-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. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.3

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

   7. Simplified 36.3%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 99.5%

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

      1. lift-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 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

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

   7. Simplified 99.5%

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

   if 1.9999999999999999e-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))))) < 0.99811092769088416

   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-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-mult N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. count-2 N/A

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

      15. cos-diff N/A

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

      16. cos-sin-sum N/A

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

      17. lower--.f64 N/A

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

      18. count-2 N/A

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

      19. lower-cos.f64 N/A

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

      20. lower-+.f64 99.4

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

      21. lift-pow.f64 N/A

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Simplified 66.5%

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, kx \cdot kx, kx\right)\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9981109276908842:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(ky \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, kx \cdot kx, kx\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-12)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx))))
   (*
    (sin th)
    (*
     (sin ky)
     (sqrt
      (/
       1.0
       (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky)))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 1e-12) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 1e-12)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	else
		tmp = Float64(sin(th) * 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)))))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 1e-12], 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]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $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]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 9.9999999999999998e-13

   1. Initial program 87.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Simplified 99.9%

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

   if 9.9999999999999998e-13 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \left(\sin ky \cdot \sin th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-12)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx))))
   (*
    (sqrt
     (/
      1.0
      (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky)))))))
    (* (sin ky) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 1e-12) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	} else {
		tmp = sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))))) * (sin(ky) * sin(th));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 1e-12)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	else
		tmp = Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky))))))) * Float64(sin(ky) * sin(th)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 1e-12], 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]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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] * N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 9.9999999999999998e-13

   1. Initial program 87.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Simplified 99.9%

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

   if 9.9999999999999998e-13 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \left(\sin ky \cdot \sin th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + \cos \left(kx + kx\right) \cdot -0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-12)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx))))
   (*
    (sin th)
    (/
     (sin ky)
     (sqrt
      (fma (- 1.0 (cos (+ ky ky))) 0.5 (+ 0.5 (* (cos (+ kx kx)) -0.5))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 1e-12) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, (0.5 + (cos((kx + kx)) * -0.5)))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 1e-12)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(0.5 + Float64(cos(Float64(kx + kx)) * -0.5))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 1e-12], 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]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(N[Cos[N[(kx + kx), $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)) < 9.9999999999999998e-13

   1. Initial program 87.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Simplified 99.9%

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

   if 9.9999999999999998e-13 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

      1. lift-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-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-mult N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. count-2 N/A

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

      15. cos-diff N/A

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

      16. cos-sin-sum N/A

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

      17. lower--.f64 N/A

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

      18. count-2 N/A

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

      19. lower-cos.f64 N/A

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

      20. lower-+.f64 99.5

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

      21. lift-pow.f64 N/A

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + \cos \left(kx + kx\right) \cdot -0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5\right) - \cos \left(kx + kx\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-12)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx))))
   (*
    (sin th)
    (/
     (sin ky)
     (sqrt
      (- (fma (- 1.0 (cos (+ ky ky))) 0.5 0.5) (* (cos (+ kx kx)) 0.5)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 1e-12) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt((fma((1.0 - cos((ky + ky))), 0.5, 0.5) - (cos((kx + kx)) * 0.5))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 1e-12)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, 0.5) - Float64(cos(Float64(kx + kx)) * 0.5)))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 1e-12], 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]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 9.9999999999999998e-13

   1. Initial program 87.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Simplified 99.9%

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

   if 9.9999999999999998e-13 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

      1. lift-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

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

      7. unpow2 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sqr-sin-a N/A

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

      11. associate-+r- N/A

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

      12. lower--.f64 N/A

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5\right) - \cos \left(kx + kx\right) \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5\right) - 0.5 \cdot \cos \left(ky + ky\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-12)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx))))
   (*
    (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) <= 1e-12) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	} 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) <= 1e-12)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, 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], 1e-12], 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]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] - N[(0.5 * N[Cos[N[(ky + ky), $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)) < 9.9999999999999998e-13

   1. Initial program 87.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Simplified 99.9%

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

   if 9.9999999999999998e-13 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

      1. lift-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. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sqr-sin-a N/A

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

      8. associate-+r- N/A

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

      9. lower--.f64 N/A

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

   4. Applied egg-rr 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5\right) - 0.5 \cdot \cos \left(ky + ky\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{1}{kx \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{\left(kx \cdot kx\right) \cdot \sqrt{0.5}}{ky \cdot \sqrt{2}}, \sqrt{0.5} \cdot \frac{\sqrt{2}}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-12)
   (*
    (sin th)
    (/
     1.0
     (*
      kx
      (fma
       -0.3333333333333333
       (/ (* (* kx kx) (sqrt 0.5)) (* ky (sqrt 2.0)))
       (* (sqrt 0.5) (/ (sqrt 2.0) ky))))))
   (* ky (/ (sin th) (sqrt (* 0.5 (- 1.0 (cos (* kx -2.0)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 1e-12) {
		tmp = sin(th) * (1.0 / (kx * fma(-0.3333333333333333, (((kx * kx) * sqrt(0.5)) / (ky * sqrt(2.0))), (sqrt(0.5) * (sqrt(2.0) / ky)))));
	} else {
		tmp = ky * (sin(th) / sqrt((0.5 * (1.0 - cos((kx * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 1e-12)
		tmp = Float64(sin(th) * Float64(1.0 / Float64(kx * fma(-0.3333333333333333, Float64(Float64(Float64(kx * kx) * sqrt(0.5)) / Float64(ky * sqrt(2.0))), Float64(sqrt(0.5) * Float64(sqrt(2.0) / ky))))));
	else
		tmp = Float64(ky * Float64(sin(th) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(kx * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 1e-12], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(kx * N[(-0.3333333333333333 * N[(N[(N[(kx * kx), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $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)) < 9.9999999999999998e-13

   1. Initial program 87.3%

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

   4. Applied egg-rr 56.2%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 2.6

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

   7. Simplified 2.6%

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

   8. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 20.0

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

   10. Simplified 20.0%

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

   if 9.9999999999999998e-13 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 52.9

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

   7. Simplified 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r/ N/A

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

   9. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx \cdot -2\right)\right) \cdot 0.5}} \cdot ky} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{1}{kx \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{\left(kx \cdot kx\right) \cdot \sqrt{0.5}}{ky \cdot \sqrt{2}}, \sqrt{0.5} \cdot \frac{\sqrt{2}}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{1}{kx \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{\left(kx \cdot kx\right) \cdot \sqrt{0.5}}{ky \cdot \sqrt{2}}, \sqrt{0.5} \cdot \frac{\sqrt{2}}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-12)
   (*
    (sin th)
    (/
     1.0
     (*
      kx
      (fma
       -0.3333333333333333
       (/ (* (* kx kx) (sqrt 0.5)) (* ky (sqrt 2.0)))
       (* (sqrt 0.5) (/ (sqrt 2.0) ky))))))
   (* (sin th) (/ ky (sqrt (* 0.5 (- 1.0 (cos (* kx -2.0)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 1e-12) {
		tmp = sin(th) * (1.0 / (kx * fma(-0.3333333333333333, (((kx * kx) * sqrt(0.5)) / (ky * sqrt(2.0))), (sqrt(0.5) * (sqrt(2.0) / ky)))));
	} else {
		tmp = sin(th) * (ky / sqrt((0.5 * (1.0 - cos((kx * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 1e-12)
		tmp = Float64(sin(th) * Float64(1.0 / Float64(kx * fma(-0.3333333333333333, Float64(Float64(Float64(kx * kx) * sqrt(0.5)) / Float64(ky * sqrt(2.0))), Float64(sqrt(0.5) * Float64(sqrt(2.0) / ky))))));
	else
		tmp = Float64(sin(th) * Float64(ky / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(kx * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 1e-12], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(kx * N[(-0.3333333333333333 * N[(N[(N[(kx * kx), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $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)) < 9.9999999999999998e-13

   1. Initial program 87.3%

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

   4. Applied egg-rr 56.2%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 2.6

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

   7. Simplified 2.6%

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

   8. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 20.0

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

   10. Simplified 20.0%

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

   if 9.9999999999999998e-13 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 52.9

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

   7. Simplified 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*l/ N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 52.0

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

   9. Applied egg-rr 52.0%

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

       1. lift-*.f64 N/A

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

       2. lift-cos.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-sin.f64 N/A

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

       8. associate-/r/ N/A

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

       9. lift-/.f64 N/A

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

       10. clear-num N/A

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

       11. lower-*.f64 N/A

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

   11. Applied egg-rr 52.9%

       \[\leadsto \color{blue}{\frac{ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}} \cdot \sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{1}{kx \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{\left(kx \cdot kx\right) \cdot \sqrt{0.5}}{ky \cdot \sqrt{2}}, \sqrt{0.5} \cdot \frac{\sqrt{2}}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.86)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx))))
   (*
    (sin th)
    (/ 1.0 (/ (* (sqrt (- 1.0 (cos (* kx -2.0)))) (sqrt 0.5)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.86) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	} else {
		tmp = sin(th) * (1.0 / ((sqrt((1.0 - cos((kx * -2.0)))) * sqrt(0.5)) / sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.86)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	else
		tmp = Float64(sin(th) * Float64(1.0 / Float64(Float64(sqrt(Float64(1.0 - cos(Float64(kx * -2.0)))) * sqrt(0.5)) / sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.86], 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]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.859999999999999987

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.8

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

   4. Applied egg-rr 99.8%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 73.9

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

   7. Simplified 73.9%

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

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-inverses N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{\color{blue}{0}}{ky - ky}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}}{\sin ky}} \cdot \sin th \]

      12. +-inverses N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}}{\sin ky}} \cdot \sin th \]

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lift-+.f64 N/A

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

      17. lower-fma.f64 33.7

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 66.7

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

   9. Simplified 66.7%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}}{\sin ky}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, kx \cdot kx, kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.86)
   (* (sin th) (/ (sin ky) (hypot (sin ky) (fma kx (* kx kx) kx))))
   (*
    (sin th)
    (/ 1.0 (/ (* (sqrt (- 1.0 (cos (* kx -2.0)))) (sqrt 0.5)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.86) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (kx * kx), kx)));
	} else {
		tmp = sin(th) * (1.0 / ((sqrt((1.0 - cos((kx * -2.0)))) * sqrt(0.5)) / sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.86)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(kx * kx), kx))));
	else
		tmp = Float64(sin(th) * Float64(1.0 / Float64(Float64(sqrt(Float64(1.0 - cos(Float64(kx * -2.0)))) * sqrt(0.5)) / sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.86], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(kx * kx), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.859999999999999987

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.8

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

   4. Applied egg-rr 99.8%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 73.9

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

   7. Simplified 73.9%

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

   8. Taylor expanded in kx around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 73.7

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

   10. Simplified 73.7%

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

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-inverses N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{\color{blue}{0}}{ky - ky}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}}{\sin ky}} \cdot \sin th \]

      12. +-inverses N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}}{\sin ky}} \cdot \sin th \]

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lift-+.f64 N/A

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

      17. lower-fma.f64 33.7

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 66.7

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

   9. Simplified 66.7%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}}{\sin ky}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, kx \cdot kx, kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 21.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{1}{kx \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{\left(kx \cdot kx\right) \cdot \sqrt{0.5}}{ky \cdot \sqrt{2}}, \sqrt{0.5} \cdot \frac{\sqrt{2}}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 0.01)
   (*
    (sin th)
    (/
     1.0
     (*
      kx
      (fma
       -0.3333333333333333
       (/ (* (* kx kx) (sqrt 0.5)) (* ky (sqrt 2.0)))
       (* (sqrt 0.5) (/ (sqrt 2.0) ky))))))
   (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (/ th (sqrt 0.5))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 0.01) {
		tmp = sin(th) * (1.0 / (kx * fma(-0.3333333333333333, (((kx * kx) * sqrt(0.5)) / (ky * sqrt(2.0))), (sqrt(0.5) * (sqrt(2.0) / ky)))));
	} else {
		tmp = sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * (th / sqrt(0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 0.01)
		tmp = Float64(sin(th) * Float64(1.0 / Float64(kx * fma(-0.3333333333333333, Float64(Float64(Float64(kx * kx) * sqrt(0.5)) / Float64(ky * sqrt(2.0))), Float64(sqrt(0.5) * Float64(sqrt(2.0) / ky))))));
	else
		tmp = Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * Float64(th / sqrt(0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(kx * N[(-0.3333333333333333 * N[(N[(N[(kx * kx), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(th / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.0100000000000000002

   1. Initial program 87.7%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 4.1

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

   7. Simplified 4.1%

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

   8. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sqrt.f64 19.6

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

   10. Simplified 19.6%

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

   if 0.0100000000000000002 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 53.0

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

   7. Simplified 53.0%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. cos-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

   10. Simplified 31.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{1}{kx \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{\left(kx \cdot kx\right) \cdot \sqrt{0.5}}{ky \cdot \sqrt{2}}, \sqrt{0.5} \cdot \frac{\sqrt{2}}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, 0.044444444444444446, -0.3333333333333333\right), 1\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.86)
   (*
    (sin th)
    (/
     1.0
     (/
      (sqrt
       (fma
        (* kx kx)
        (fma
         (* kx kx)
         (fma (* kx kx) 0.044444444444444446 -0.3333333333333333)
         1.0)
        (fma -0.5 (cos (* ky -2.0)) 0.5)))
      (sin ky))))
   (*
    (sin th)
    (/ 1.0 (/ (* (sqrt (- 1.0 (cos (* kx -2.0)))) (sqrt 0.5)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.86) {
		tmp = sin(th) * (1.0 / (sqrt(fma((kx * kx), fma((kx * kx), fma((kx * kx), 0.044444444444444446, -0.3333333333333333), 1.0), fma(-0.5, cos((ky * -2.0)), 0.5))) / sin(ky)));
	} else {
		tmp = sin(th) * (1.0 / ((sqrt((1.0 - cos((kx * -2.0)))) * sqrt(0.5)) / sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.86)
		tmp = Float64(sin(th) * Float64(1.0 / Float64(sqrt(fma(Float64(kx * kx), fma(Float64(kx * kx), fma(Float64(kx * kx), 0.044444444444444446, -0.3333333333333333), 1.0), fma(-0.5, cos(Float64(ky * -2.0)), 0.5))) / sin(ky))));
	else
		tmp = Float64(sin(th) * Float64(1.0 / Float64(Float64(sqrt(Float64(1.0 - cos(Float64(kx * -2.0)))) * sqrt(0.5)) / sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.86], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(N[Sqrt[N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * 0.044444444444444446 + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] + N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.859999999999999987

   1. Initial program 91.4%

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

   4. Applied egg-rr 70.6%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}}{\sin ky}} \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right) + \frac{1}{2}}}}{\sin ky}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\left({kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \frac{1}{2}}}{\sin ky}} \cdot \sin th \]

      3. associate-+l+ N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}\right)}}}{\sin ky}} \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) + \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left({kx}^{2}, 1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{kx \cdot kx}, 1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      8. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) + 1}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, 1\right)}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, 1\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, 1\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      12. sub-neg N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{2}{45} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{2}{45}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), 1\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, {kx}^{2} \cdot \frac{2}{45} + \color{blue}{\frac{-1}{3}}, 1\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{2}{45}, \frac{-1}{3}\right)}, 1\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      16. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45}, \frac{-1}{3}\right), 1\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45}, \frac{-1}{3}\right), 1\right), \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}{\sin ky}} \cdot \sin th \]

   7. Simplified 58.1%

      \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, 0.044444444444444446, -0.3333333333333333\right), 1\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}}{\sin ky}} \cdot \sin th \]

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-inverses N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{\color{blue}{0}}{ky - ky}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}}{\sin ky}} \cdot \sin th \]

      12. +-inverses N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}}{\sin ky}} \cdot \sin th \]

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lift-+.f64 N/A

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

      17. lower-fma.f64 33.7

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 66.7

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

   9. Simplified 66.7%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}}{\sin ky}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, 0.044444444444444446, -0.3333333333333333\right), 1\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 21.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.01:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{2} \cdot \left(kx \cdot \sqrt{0.5}\right)}{ky}}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 0.01)
   (/ 1.0 (/ (/ (* (sqrt 2.0) (* kx (sqrt 0.5))) ky) (sin th)))
   (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (/ th (sqrt 0.5))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 0.01) {
		tmp = 1.0 / (((sqrt(2.0) * (kx * sqrt(0.5))) / ky) / sin(th));
	} else {
		tmp = sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * (th / sqrt(0.5)));
	}
	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(kx) ** 2.0d0) <= 0.01d0) then
        tmp = 1.0d0 / (((sqrt(2.0d0) * (kx * sqrt(0.5d0))) / ky) / sin(th))
    else
        tmp = sqrt((1.0d0 / (1.0d0 - cos((kx * (-2.0d0)))))) * (ky * (th / sqrt(0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(kx), 2.0) <= 0.01) {
		tmp = 1.0 / (((Math.sqrt(2.0) * (kx * Math.sqrt(0.5))) / ky) / Math.sin(th));
	} else {
		tmp = Math.sqrt((1.0 / (1.0 - Math.cos((kx * -2.0))))) * (ky * (th / Math.sqrt(0.5)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(kx), 2.0) <= 0.01:
		tmp = 1.0 / (((math.sqrt(2.0) * (kx * math.sqrt(0.5))) / ky) / math.sin(th))
	else:
		tmp = math.sqrt((1.0 / (1.0 - math.cos((kx * -2.0))))) * (ky * (th / math.sqrt(0.5)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 0.01)
		tmp = Float64(1.0 / Float64(Float64(Float64(sqrt(2.0) * Float64(kx * sqrt(0.5))) / ky) / sin(th)));
	else
		tmp = Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * Float64(th / sqrt(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(kx) ^ 2.0) <= 0.01)
		tmp = 1.0 / (((sqrt(2.0) * (kx * sqrt(0.5))) / ky) / sin(th));
	else
		tmp = sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * (th / sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 0.01], N[(1.0 / N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(kx * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(th / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.0100000000000000002

   1. Initial program 87.7%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 4.1

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

   7. Simplified 4.1%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*l/ N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 4.1

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

   9. Applied egg-rr 4.1%

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

   10. Taylor expanded in kx around 0

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

       1. lower-/.f64 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-sqrt.f64 19.5

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

   12. Simplified 19.5%

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

   if 0.0100000000000000002 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 53.0

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

   7. Simplified 53.0%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. cos-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

   10. Simplified 31.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.01:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{2} \cdot \left(kx \cdot \sqrt{0.5}\right)}{ky}}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(kx \cdot -2\right)\\ \mathbf{if}\;ky \leq 3.5 \cdot 10^{-183}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, {ky}^{6}, -0.16666666666666666\right), ky\right)}{kx}\\ \mathbf{elif}\;ky \leq 3.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\sqrt{1 - t\_1} \cdot \frac{\sqrt{0.5}}{ky \cdot \sin th}}\\ \mathbf{elif}\;ky \leq 2.9 \cdot 10^{-92}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;ky \leq 0.0048:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, t\_1, \mathsf{fma}\left(ky, ky, 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* kx -2.0))))
   (if (<= ky 3.5e-183)
     (*
      (sin th)
      (/
       (fma
        ky
        (*
         (pow ky 6.0)
         (fma 0.008333333333333333 (pow ky 6.0) -0.16666666666666666))
        ky)
       kx))
     (if (<= ky 3.5e-161)
       (/ 1.0 (* (sqrt (- 1.0 t_1)) (/ (sqrt 0.5) (* ky (sin th)))))
       (if (<= ky 2.9e-92)
         (* (sin th) (/ (sin ky) (sqrt (fma (* 2.0 (* ky ky)) 0.5 (* kx kx)))))
         (if (<= ky 0.0048)
           (* (sin th) (/ (sin ky) (sqrt (fma -0.5 t_1 (fma ky ky 0.5)))))
           (*
            (sin th)
            (/ (sin ky) (sqrt (* 0.5 (- 1.0 (cos (* ky -2.0)))))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((kx * -2.0));
	double tmp;
	if (ky <= 3.5e-183) {
		tmp = sin(th) * (fma(ky, (pow(ky, 6.0) * fma(0.008333333333333333, pow(ky, 6.0), -0.16666666666666666)), ky) / kx);
	} else if (ky <= 3.5e-161) {
		tmp = 1.0 / (sqrt((1.0 - t_1)) * (sqrt(0.5) / (ky * sin(th))));
	} else if (ky <= 2.9e-92) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((2.0 * (ky * ky)), 0.5, (kx * kx))));
	} else if (ky <= 0.0048) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, t_1, fma(ky, ky, 0.5))));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt((0.5 * (1.0 - cos((ky * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(kx * -2.0))
	tmp = 0.0
	if (ky <= 3.5e-183)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64((ky ^ 6.0) * fma(0.008333333333333333, (ky ^ 6.0), -0.16666666666666666)), ky) / kx));
	elseif (ky <= 3.5e-161)
		tmp = Float64(1.0 / Float64(sqrt(Float64(1.0 - t_1)) * Float64(sqrt(0.5) / Float64(ky * sin(th)))));
	elseif (ky <= 2.9e-92)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(2.0 * Float64(ky * ky)), 0.5, Float64(kx * kx)))));
	elseif (ky <= 0.0048)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, t_1, fma(ky, ky, 0.5)))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(ky * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ky, 3.5e-183], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[Power[ky, 6.0], $MachinePrecision] * N[(0.008333333333333333 * N[Power[ky, 6.0], $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 3.5e-161], N[(1.0 / N[(N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 2.9e-92], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 0.0048], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * t$95$1 + N[(ky * ky + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if ky < 3.49999999999999991e-183

   1. Initial program 91.5%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)\right)} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)}\right) \cdot \sin th \]

      3. distribute-lft-in N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right) + {ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)} + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) \cdot \sin th \]

      4. associate-+l+ N/A

         \[\leadsto \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right) + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)}\right) \cdot \sin th \]

      5. associate-*r* N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \frac{-1}{6}\right) \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}} + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2}\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right) \cdot \sin th \]

   5. Simplified 8.4%

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right)} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      2. lower-/.f64 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\color{blue}{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 13.3

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

   8. Simplified 13.3%

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

   9. Taylor expanded in kx around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. *-rgt-identity N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-pow.f64 25.3

           \[\leadsto \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{{ky}^{6}}, -0.16666666666666666\right), ky\right)}{kx} \cdot \sin th \]

   11. Simplified 25.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, {ky}^{6}, -0.16666666666666666\right), ky\right)}{kx}} \cdot \sin th \]

   if 3.49999999999999991e-183 < ky < 3.5000000000000002e-161

   1. Initial program 75.3%

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

   4. Applied egg-rr 62.9%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 63.0

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

   7. Simplified 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*l/ N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 63.0

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

   9. Applied egg-rr 62.9%

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

   10. Taylor expanded in kx around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. cos-neg N/A

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

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

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

       8. metadata-eval N/A

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

       9. lower-sqrt.f64 N/A

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

       10. metadata-eval N/A

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

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

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

       12. lower--.f64 N/A

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

       13. cos-neg N/A

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

       14. lower-cos.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 63.1

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

   12. Simplified 63.1%

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

   if 3.5000000000000002e-161 < ky < 2.89999999999999985e-92

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-mult N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. count-2 N/A

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

      15. cos-diff N/A

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

      16. cos-sin-sum N/A

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

      17. lower--.f64 N/A

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

      18. count-2 N/A

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

      19. lower-cos.f64 N/A

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

      20. lower-+.f64 64.6

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

      21. lift-pow.f64 N/A

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

   4. Applied egg-rr 55.5%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 38.9

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

   7. Simplified 38.9%

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

   8. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 73.9

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

   10. Simplified 73.9%

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

   if 2.89999999999999985e-92 < ky < 0.00479999999999999958

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-mult N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. count-2 N/A

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

      15. cos-diff N/A

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

      16. cos-sin-sum N/A

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

      17. lower--.f64 N/A

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

      18. count-2 N/A

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

      19. lower-cos.f64 N/A

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

      20. lower-+.f64 73.9

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

      21. lift-pow.f64 N/A

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

   4. Applied egg-rr 72.9%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 72.9

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

   7. Simplified 72.9%

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

   if 0.00479999999999999958 < ky

   1. Initial program 99.7%

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

      1. lift-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-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-mult N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. count-2 N/A

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

      15. cos-diff N/A

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

      16. cos-sin-sum N/A

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

      17. lower--.f64 N/A

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

      18. count-2 N/A

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

      19. lower-cos.f64 N/A

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

      20. lower-+.f64 99.5

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

      21. lift-pow.f64 N/A

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower--.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 55.4

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

   7. Simplified 55.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.5 \cdot 10^{-183}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, {ky}^{6}, -0.16666666666666666\right), ky\right)}{kx}\\ \mathbf{elif}\;ky \leq 3.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky \cdot \sin th}}\\ \mathbf{elif}\;ky \leq 2.9 \cdot 10^{-92}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;ky \leq 0.0048:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), \mathsf{fma}\left(ky, ky, 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 21.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{2} \cdot \left(kx \cdot \sqrt{0.5}\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 0.01)
   (* (sin th) (/ 1.0 (/ (* (sqrt 2.0) (* kx (sqrt 0.5))) ky)))
   (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (/ th (sqrt 0.5))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 0.01) {
		tmp = sin(th) * (1.0 / ((sqrt(2.0) * (kx * sqrt(0.5))) / ky));
	} else {
		tmp = sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * (th / sqrt(0.5)));
	}
	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(kx) ** 2.0d0) <= 0.01d0) then
        tmp = sin(th) * (1.0d0 / ((sqrt(2.0d0) * (kx * sqrt(0.5d0))) / ky))
    else
        tmp = sqrt((1.0d0 / (1.0d0 - cos((kx * (-2.0d0)))))) * (ky * (th / sqrt(0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(kx), 2.0) <= 0.01) {
		tmp = Math.sin(th) * (1.0 / ((Math.sqrt(2.0) * (kx * Math.sqrt(0.5))) / ky));
	} else {
		tmp = Math.sqrt((1.0 / (1.0 - Math.cos((kx * -2.0))))) * (ky * (th / Math.sqrt(0.5)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(kx), 2.0) <= 0.01:
		tmp = math.sin(th) * (1.0 / ((math.sqrt(2.0) * (kx * math.sqrt(0.5))) / ky))
	else:
		tmp = math.sqrt((1.0 / (1.0 - math.cos((kx * -2.0))))) * (ky * (th / math.sqrt(0.5)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 0.01)
		tmp = Float64(sin(th) * Float64(1.0 / Float64(Float64(sqrt(2.0) * Float64(kx * sqrt(0.5))) / ky)));
	else
		tmp = Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * Float64(th / sqrt(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(kx) ^ 2.0) <= 0.01)
		tmp = sin(th) * (1.0 / ((sqrt(2.0) * (kx * sqrt(0.5))) / ky));
	else
		tmp = sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * (th / sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(kx * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(th / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.0100000000000000002

   1. Initial program 87.7%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 4.1

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

   7. Simplified 4.1%

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

   8. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 19.5

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

   10. Simplified 19.5%

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

   if 0.0100000000000000002 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 53.0

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

   7. Simplified 53.0%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. cos-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

   10. Simplified 31.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{2} \cdot \left(kx \cdot \sqrt{0.5}\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.86)
   (* (sin th) (/ (sin ky) (sqrt (fma (- 1.0 (cos (+ ky ky))) 0.5 (* kx kx)))))
   (*
    (sin th)
    (/ 1.0 (/ (* (sqrt (- 1.0 (cos (* kx -2.0)))) (sqrt 0.5)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.86) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, (kx * kx))));
	} else {
		tmp = sin(th) * (1.0 / ((sqrt((1.0 - cos((kx * -2.0)))) * sqrt(0.5)) / sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.86)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(kx * kx)))));
	else
		tmp = Float64(sin(th) * Float64(1.0 / Float64(Float64(sqrt(Float64(1.0 - cos(Float64(kx * -2.0)))) * sqrt(0.5)) / sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.86], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.859999999999999987

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. sin-mult N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. count-2 N/A

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

      15. cos-diff N/A

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

      16. cos-sin-sum N/A

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

      17. lower--.f64 N/A

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

      18. count-2 N/A

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

      19. lower-cos.f64 N/A

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

      20. lower-+.f64 83.9

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

      21. lift-pow.f64 N/A

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

   4. Applied egg-rr 70.7%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.2

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

   7. Simplified 58.2%

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

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-inverses N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{\color{blue}{0}}{ky - ky}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}}{\sin ky}} \cdot \sin th \]

      12. +-inverses N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}}{\sin ky}} \cdot \sin th \]

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lift-+.f64 N/A

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

      17. lower-fma.f64 33.7

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 66.7

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

   9. Simplified 66.7%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}}{\sin ky}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 21.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{2} \cdot \left(kx \cdot \sqrt{0.5}\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 0.01)
   (* (sin th) (/ 1.0 (/ (* (sqrt 2.0) (* kx (sqrt 0.5))) ky)))
   (*
    (fma th (* -0.16666666666666666 (* th th)) th)
    (* ky (sqrt (/ 1.0 (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) <= 0.01) {
		tmp = sin(th) * (1.0 / ((sqrt(2.0) * (kx * sqrt(0.5))) / ky));
	} else {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (ky * sqrt((1.0 / 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) <= 0.01)
		tmp = Float64(sin(th) * Float64(1.0 / Float64(Float64(sqrt(2.0) * Float64(kx * sqrt(0.5))) / ky)));
	else
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(ky * sqrt(Float64(1.0 / 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], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(kx * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(ky * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.0100000000000000002

   1. Initial program 87.7%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 4.1

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

   7. Simplified 4.1%

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

   8. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(kx \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{2}}{ky}} \cdot \sin th \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(kx \cdot \color{blue}{\sqrt{\frac{1}{2}}}\right) \cdot \sqrt{2}}{ky}} \cdot \sin th \]

      6. lower-sqrt.f64 19.5

         \[\leadsto \frac{1}{\frac{\left(kx \cdot \sqrt{0.5}\right) \cdot \color{blue}{\sqrt{2}}}{ky}} \cdot \sin th \]

   10. Simplified 19.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(kx \cdot \sqrt{0.5}\right) \cdot \sqrt{2}}{ky}}} \cdot \sin th \]

   if 0.0100000000000000002 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 99.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 53.8

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   7. Simplified 53.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}}\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(ky \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}}\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      4. +-commutative N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2}}}}\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}}\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}}\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}}\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      8. cos-neg N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      10. *-commutative N/A

          \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      11. lower-*.f64 30.5

          \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   10. Simplified 30.5%

       \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{1}{\frac{\sqrt{2} \cdot \left(kx \cdot \sqrt{0.5}\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \sin th\\ t_2 := 1 - \cos \left(kx \cdot -2\right)\\ \mathbf{if}\;ky \leq 3.5 \cdot 10^{-183}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, {ky}^{6}, -0.16666666666666666\right), ky\right)}{kx}\\ \mathbf{elif}\;ky \leq 3.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\sqrt{t\_2} \cdot \frac{\sqrt{0.5}}{t\_1}}\\ \mathbf{elif}\;ky \leq 2.9 \cdot 10^{-92}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;ky \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{1}{t\_2}} \cdot \left(t\_1 \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (sin th))) (t_2 (- 1.0 (cos (* kx -2.0)))))
   (if (<= ky 3.5e-183)
     (*
      (sin th)
      (/
       (fma
        ky
        (*
         (pow ky 6.0)
         (fma 0.008333333333333333 (pow ky 6.0) -0.16666666666666666))
        ky)
       kx))
     (if (<= ky 3.5e-161)
       (/ 1.0 (* (sqrt t_2) (/ (sqrt 0.5) t_1)))
       (if (<= ky 2.9e-92)
         (* (sin th) (/ (sin ky) (sqrt (fma (* 2.0 (* ky ky)) 0.5 (* kx kx)))))
         (if (<= ky 8.5e-5)
           (* (sqrt (/ 1.0 t_2)) (* t_1 (sqrt 2.0)))
           (*
            (sin th)
            (/ (sin ky) (sqrt (* 0.5 (- 1.0 (cos (* ky -2.0)))))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * sin(th);
	double t_2 = 1.0 - cos((kx * -2.0));
	double tmp;
	if (ky <= 3.5e-183) {
		tmp = sin(th) * (fma(ky, (pow(ky, 6.0) * fma(0.008333333333333333, pow(ky, 6.0), -0.16666666666666666)), ky) / kx);
	} else if (ky <= 3.5e-161) {
		tmp = 1.0 / (sqrt(t_2) * (sqrt(0.5) / t_1));
	} else if (ky <= 2.9e-92) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((2.0 * (ky * ky)), 0.5, (kx * kx))));
	} else if (ky <= 8.5e-5) {
		tmp = sqrt((1.0 / t_2)) * (t_1 * sqrt(2.0));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt((0.5 * (1.0 - cos((ky * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * sin(th))
	t_2 = Float64(1.0 - cos(Float64(kx * -2.0)))
	tmp = 0.0
	if (ky <= 3.5e-183)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64((ky ^ 6.0) * fma(0.008333333333333333, (ky ^ 6.0), -0.16666666666666666)), ky) / kx));
	elseif (ky <= 3.5e-161)
		tmp = Float64(1.0 / Float64(sqrt(t_2) * Float64(sqrt(0.5) / t_1)));
	elseif (ky <= 2.9e-92)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(2.0 * Float64(ky * ky)), 0.5, Float64(kx * kx)))));
	elseif (ky <= 8.5e-5)
		tmp = Float64(sqrt(Float64(1.0 / t_2)) * Float64(t_1 * sqrt(2.0)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(ky * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ky, 3.5e-183], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[Power[ky, 6.0], $MachinePrecision] * N[(0.008333333333333333 * N[Power[ky, 6.0], $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 3.5e-161], N[(1.0 / N[(N[Sqrt[t$95$2], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 2.9e-92], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 8.5e-5], N[(N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if ky < 3.49999999999999991e-183

   1. Initial program 91.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)\right)} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)}\right) \cdot \sin th \]

      3. distribute-lft-in N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right) + {ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)} + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) \cdot \sin th \]

      4. associate-+l+ N/A

         \[\leadsto \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right) + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)}\right) \cdot \sin th \]

      5. associate-*r* N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \frac{-1}{6}\right) \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}} + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2}\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right) \cdot \sin th \]

   5. Simplified 8.4%

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right)} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      2. lower-/.f64 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\color{blue}{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      3. cube-mult N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{\color{blue}{kx \cdot \left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{{kx}^{2}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{\color{blue}{kx \cdot {kx}^{2}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{\left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      7. lower-*.f64 13.3

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{\left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

   8. Simplified 13.3%

      \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{kx \cdot \left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

   9. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\frac{ky \cdot \left(1 + {ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right)\right)}{kx}} \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{ky \cdot \left(1 + {ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right)\right)}{kx}} \cdot \sin th \]

       2. +-commutative N/A

          \[\leadsto \frac{ky \cdot \color{blue}{\left({ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right) + 1\right)}}{kx} \cdot \sin th \]

       3. distribute-lft-in N/A

          \[\leadsto \frac{\color{blue}{ky \cdot \left({ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right)\right) + ky \cdot 1}}{kx} \cdot \sin th \]

       4. *-rgt-identity N/A

          \[\leadsto \frac{ky \cdot \left({ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right)\right) + \color{blue}{ky}}{kx} \cdot \sin th \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, {ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right), ky\right)}}{kx} \cdot \sin th \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right)}, ky\right)}{kx} \cdot \sin th \]

       7. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{6}} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right), ky\right)}{kx} \cdot \sin th \]

       8. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \color{blue}{\left(\frac{1}{120} \cdot {ky}^{6} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, ky\right)}{kx} \cdot \sin th \]

       9. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} + \color{blue}{\frac{-1}{6}}\right), ky\right)}{kx} \cdot \sin th \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{6}, \frac{-1}{6}\right)}, ky\right)}{kx} \cdot \sin th \]

       11. lower-pow.f64 25.3

           \[\leadsto \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{{ky}^{6}}, -0.16666666666666666\right), ky\right)}{kx} \cdot \sin th \]

   11. Simplified 25.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, {ky}^{6}, -0.16666666666666666\right), ky\right)}{kx}} \cdot \sin th \]

   if 3.49999999999999991e-183 < ky < 3.5000000000000002e-161

   1. Initial program 75.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      7. cos-neg N/A

         \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      12. lower-sqrt.f64 63.0

          \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\color{blue}{\sqrt{0.5}}}{ky}} \cdot \sin th \]

   7. Simplified 63.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      3. lift--.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\color{blue}{1 - \cos \left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\color{blue}{\sqrt{\frac{1}{2}}}}{ky}} \cdot \sin th \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \color{blue}{\sin th} \]

      9. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}{1 \cdot \sin th}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}{1 \cdot \sin th}}} \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}{\color{blue}{\sin th}}} \]

      13. lower-/.f64 63.0

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky}}{\sin th}}} \]

   9. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sqrt{\left(1 - \cos \left(kx \cdot -2\right)\right) \cdot 0.5}}{ky}}{\sin th}}} \]

   10. Taylor expanded in kx around inf

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th}} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{2}}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{\color{blue}{ky \cdot \sin th}} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}} \]

       5. lower-sin.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \color{blue}{\sin th}} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}} \]

       6. cos-neg N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot kx\right)}}} \]

       8. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(\color{blue}{2} \cdot kx\right)}} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \]

       11. distribute-lft-neg-in N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \]

       12. lower--.f64 N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \]

       13. cos-neg N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \]

       14. lower-cos.f64 N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \]

       15. *-commutative N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \]

       16. lower-*.f64 63.1

           \[\leadsto \frac{1}{\frac{\sqrt{0.5}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \]

   12. Simplified 63.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{0.5}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(kx \cdot -2\right)}}} \]

   if 3.5000000000000002e-161 < ky < 2.89999999999999985e-92

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 64.6

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 55.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 38.9

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   7. Simplified 38.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      3. lower-*.f64 73.9

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   10. Simplified 73.9%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot \left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   if 2.89999999999999985e-92 < ky < 8.500000000000001e-5

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      8. cos-neg N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\color{blue}{\left(ky \cdot \sin th\right)} \cdot \sqrt{2}\right) \]

      15. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\left(ky \cdot \color{blue}{\sin th}\right) \cdot \sqrt{2}\right) \]

      16. lower-sqrt.f64 72.1

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\left(ky \cdot \sin th\right) \cdot \color{blue}{\sqrt{2}}\right) \]

   7. Simplified 72.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

   if 8.500000000000001e-5 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 99.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right)}} \cdot \sin th \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right)}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)}}} \cdot \sin th \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \left(1 - \color{blue}{\cos \left(-2 \cdot ky\right)}\right)}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \left(1 - \color{blue}{\cos \left(-2 \cdot ky\right)}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \color{blue}{\left(ky \cdot -2\right)}\right)}} \cdot \sin th \]

      8. lower-*.f64 55.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \color{blue}{\left(ky \cdot -2\right)}\right)}} \cdot \sin th \]

   7. Simplified 55.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.5 \cdot 10^{-183}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, {ky}^{6}, -0.16666666666666666\right), ky\right)}{kx}\\ \mathbf{elif}\;ky \leq 3.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky \cdot \sin th}}\\ \mathbf{elif}\;ky \leq 2.9 \cdot 10^{-92}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;ky \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 36.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \sin th\\ t_2 := 1 - \cos \left(kx \cdot -2\right)\\ \mathbf{if}\;ky \leq 3.5 \cdot 10^{-183}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, {ky}^{6}, -0.16666666666666666\right), ky\right)}{kx}\\ \mathbf{elif}\;ky \leq 3.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\sqrt{t\_2} \cdot \frac{\sqrt{0.5}}{t\_1}}\\ \mathbf{elif}\;ky \leq 2.9 \cdot 10^{-92}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;ky \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{1}{t\_2}} \cdot \left(t\_1 \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (sin th))) (t_2 (- 1.0 (cos (* kx -2.0)))))
   (if (<= ky 3.5e-183)
     (*
      (sin th)
      (/
       (fma
        ky
        (*
         (pow ky 6.0)
         (fma 0.008333333333333333 (pow ky 6.0) -0.16666666666666666))
        ky)
       kx))
     (if (<= ky 3.5e-161)
       (/ 1.0 (* (sqrt t_2) (/ (sqrt 0.5) t_1)))
       (if (<= ky 2.9e-92)
         (* (sin th) (/ (sin ky) (sqrt (fma (* 2.0 (* ky ky)) 0.5 (* kx kx)))))
         (if (<= ky 8.5e-5)
           (* (sqrt (/ 1.0 t_2)) (* t_1 (sqrt 2.0)))
           (*
            (sin ky)
            (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * sin(th);
	double t_2 = 1.0 - cos((kx * -2.0));
	double tmp;
	if (ky <= 3.5e-183) {
		tmp = sin(th) * (fma(ky, (pow(ky, 6.0) * fma(0.008333333333333333, pow(ky, 6.0), -0.16666666666666666)), ky) / kx);
	} else if (ky <= 3.5e-161) {
		tmp = 1.0 / (sqrt(t_2) * (sqrt(0.5) / t_1));
	} else if (ky <= 2.9e-92) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((2.0 * (ky * ky)), 0.5, (kx * kx))));
	} else if (ky <= 8.5e-5) {
		tmp = sqrt((1.0 / t_2)) * (t_1 * sqrt(2.0));
	} else {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * sin(th))
	t_2 = Float64(1.0 - cos(Float64(kx * -2.0)))
	tmp = 0.0
	if (ky <= 3.5e-183)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64((ky ^ 6.0) * fma(0.008333333333333333, (ky ^ 6.0), -0.16666666666666666)), ky) / kx));
	elseif (ky <= 3.5e-161)
		tmp = Float64(1.0 / Float64(sqrt(t_2) * Float64(sqrt(0.5) / t_1)));
	elseif (ky <= 2.9e-92)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(2.0 * Float64(ky * ky)), 0.5, Float64(kx * kx)))));
	elseif (ky <= 8.5e-5)
		tmp = Float64(sqrt(Float64(1.0 / t_2)) * Float64(t_1 * sqrt(2.0)));
	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_] := Block[{t$95$1 = N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ky, 3.5e-183], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[Power[ky, 6.0], $MachinePrecision] * N[(0.008333333333333333 * N[Power[ky, 6.0], $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 3.5e-161], N[(1.0 / N[(N[Sqrt[t$95$2], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 2.9e-92], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 8.5e-5], N[(N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $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 5 regimes

2. if ky < 3.49999999999999991e-183

   1. Initial program 91.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)\right)} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)}\right) \cdot \sin th \]

      3. distribute-lft-in N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right) + {ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)} + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) \cdot \sin th \]

      4. associate-+l+ N/A

         \[\leadsto \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right) + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)}\right) \cdot \sin th \]

      5. associate-*r* N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \frac{-1}{6}\right) \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}} + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2}\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right) \cdot \sin th \]

   5. Simplified 8.4%

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right)} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      2. lower-/.f64 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\color{blue}{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      3. cube-mult N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{\color{blue}{kx \cdot \left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{{kx}^{2}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{\color{blue}{kx \cdot {kx}^{2}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{\left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      7. lower-*.f64 13.3

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{\left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

   8. Simplified 13.3%

      \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{kx \cdot \left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

   9. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\frac{ky \cdot \left(1 + {ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right)\right)}{kx}} \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{ky \cdot \left(1 + {ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right)\right)}{kx}} \cdot \sin th \]

       2. +-commutative N/A

          \[\leadsto \frac{ky \cdot \color{blue}{\left({ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right) + 1\right)}}{kx} \cdot \sin th \]

       3. distribute-lft-in N/A

          \[\leadsto \frac{\color{blue}{ky \cdot \left({ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right)\right) + ky \cdot 1}}{kx} \cdot \sin th \]

       4. *-rgt-identity N/A

          \[\leadsto \frac{ky \cdot \left({ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right)\right) + \color{blue}{ky}}{kx} \cdot \sin th \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, {ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right), ky\right)}}{kx} \cdot \sin th \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right)}, ky\right)}{kx} \cdot \sin th \]

       7. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{6}} \cdot \left(\frac{1}{120} \cdot {ky}^{6} - \frac{1}{6}\right), ky\right)}{kx} \cdot \sin th \]

       8. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \color{blue}{\left(\frac{1}{120} \cdot {ky}^{6} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, ky\right)}{kx} \cdot \sin th \]

       9. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \left(\frac{1}{120} \cdot {ky}^{6} + \color{blue}{\frac{-1}{6}}\right), ky\right)}{kx} \cdot \sin th \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{6}, \frac{-1}{6}\right)}, ky\right)}{kx} \cdot \sin th \]

       11. lower-pow.f64 25.3

           \[\leadsto \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{{ky}^{6}}, -0.16666666666666666\right), ky\right)}{kx} \cdot \sin th \]

   11. Simplified 25.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, {ky}^{6}, -0.16666666666666666\right), ky\right)}{kx}} \cdot \sin th \]

   if 3.49999999999999991e-183 < ky < 3.5000000000000002e-161

   1. Initial program 75.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      7. cos-neg N/A

         \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      12. lower-sqrt.f64 63.0

          \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\color{blue}{\sqrt{0.5}}}{ky}} \cdot \sin th \]

   7. Simplified 63.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      3. lift--.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\color{blue}{1 - \cos \left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\color{blue}{\sqrt{\frac{1}{2}}}}{ky}} \cdot \sin th \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \color{blue}{\sin th} \]

      9. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \]

      10. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}{1 \cdot \sin th}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}{1 \cdot \sin th}}} \]

      12. *-lft-identity N/A

          \[\leadsto \frac{1}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}{\color{blue}{\sin th}}} \]

      13. lower-/.f64 63.0

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky}}{\sin th}}} \]

   9. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sqrt{\left(1 - \cos \left(kx \cdot -2\right)\right) \cdot 0.5}}{ky}}{\sin th}}} \]

   10. Taylor expanded in kx around inf

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th}} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{2}}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{\color{blue}{ky \cdot \sin th}} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}} \]

       5. lower-sin.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \color{blue}{\sin th}} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}} \]

       6. cos-neg N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot kx\right)}}} \]

       8. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(\color{blue}{2} \cdot kx\right)}} \]

       9. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \]

       11. distribute-lft-neg-in N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \]

       12. lower--.f64 N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \]

       13. cos-neg N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \]

       14. lower-cos.f64 N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \]

       15. *-commutative N/A

           \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{2}}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \]

       16. lower-*.f64 63.1

           \[\leadsto \frac{1}{\frac{\sqrt{0.5}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \]

   12. Simplified 63.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{0.5}}{ky \cdot \sin th} \cdot \sqrt{1 - \cos \left(kx \cdot -2\right)}}} \]

   if 3.5000000000000002e-161 < ky < 2.89999999999999985e-92

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 64.6

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 55.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 38.9

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   7. Simplified 38.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      3. lower-*.f64 73.9

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   10. Simplified 73.9%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot \left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   if 2.89999999999999985e-92 < ky < 8.500000000000001e-5

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      8. cos-neg N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\color{blue}{\left(ky \cdot \sin th\right)} \cdot \sqrt{2}\right) \]

      15. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\left(ky \cdot \color{blue}{\sin th}\right) \cdot \sqrt{2}\right) \]

      16. lower-sqrt.f64 72.1

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\left(ky \cdot \sin th\right) \cdot \color{blue}{\sqrt{2}}\right) \]

   7. Simplified 72.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

   if 8.500000000000001e-5 < ky

   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 99.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky} \]

   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. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right) + \frac{1}{2}}} \cdot \sin ky \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)} + \frac{1}{2}}} \cdot \sin ky \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right), \frac{1}{2}\right)}}} \cdot \sin ky \]

      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 55.3

         \[\leadsto \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \cdot \sin ky \]

   7. Simplified 55.3%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \cdot \sin ky \]
3. Recombined 5 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.5 \cdot 10^{-183}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, {ky}^{6} \cdot \mathsf{fma}\left(0.008333333333333333, {ky}^{6}, -0.16666666666666666\right), ky\right)}{kx}\\ \mathbf{elif}\;ky \leq 3.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky \cdot \sin th}}\\ \mathbf{elif}\;ky \leq 2.9 \cdot 10^{-92}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;ky \leq 8.5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 38.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;kx \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 5e-201)
   (* (sin th) (/ (sin ky) (sqrt (fma (* 2.0 (* ky ky)) 0.5 (* kx kx)))))
   (if (<= kx 0.86)
     (*
      (fma th (* -0.16666666666666666 (* th th)) th)
      (/
       (sin ky)
       (sqrt
        (fma
         (- 1.0 (cos (+ ky ky)))
         0.5
         (*
          (* kx kx)
          (fma
           (* kx kx)
           (fma
            (* kx kx)
            (fma (* kx kx) -0.0031746031746031746 0.044444444444444446)
            -0.3333333333333333)
           1.0))))))
     (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 5e-201) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((2.0 * (ky * ky)), 0.5, (kx * kx))));
	} else if (kx <= 0.86) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, ((kx * kx) * fma((kx * kx), fma((kx * kx), fma((kx * kx), -0.0031746031746031746, 0.044444444444444446), -0.3333333333333333), 1.0)))));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 5e-201)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(2.0 * Float64(ky * ky)), 0.5, Float64(kx * kx)))));
	elseif (kx <= 0.86)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(kx * kx) * fma(Float64(kx * kx), fma(Float64(kx * kx), fma(Float64(kx * kx), -0.0031746031746031746, 0.044444444444444446), -0.3333333333333333), 1.0))))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 5e-201], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 0.86], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 4.9999999999999999e-201

   1. Initial program 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 83.4

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 75.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 50.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   7. Simplified 50.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      3. lower-*.f64 27.7

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   10. Simplified 27.7%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot \left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   if 4.9999999999999999e-201 < kx < 0.859999999999999987

   1. Initial program 90.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 85.8

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 54.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 32.5

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   7. Simplified 32.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\left({kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right) + 1\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\mathsf{fma}\left({kx}^{2}, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}, 1\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) + \color{blue}{\frac{-1}{3}}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}, \frac{-1}{3}\right)}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{-1}{315} \cdot {kx}^{2} + \frac{2}{45}}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{-1}{315}} + \frac{2}{45}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{-1}{315}, \frac{2}{45}\right)}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{-1}{315}, \frac{2}{45}\right), \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      17. lower-*.f64 51.3

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   10. Simplified 51.3%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)}\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 99.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2}}}} \cdot \sin th \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \cdot \sin th \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \cdot \sin th \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 66.6

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \cdot \sin th \]

   7. Simplified 66.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;kx \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 55.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.86)
   (* (sin th) (/ (sin ky) (sqrt (fma (- 1.0 (cos (+ ky ky))) 0.5 (* kx kx)))))
   (* (sin ky) (/ (sin th) (sqrt (* 0.5 (- 1.0 (cos (* kx -2.0)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.86) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, (kx * kx))));
	} else {
		tmp = sin(ky) * (sin(th) / sqrt((0.5 * (1.0 - cos((kx * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.86)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(kx * kx)))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(kx * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.86], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.859999999999999987

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 83.9

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 70.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 58.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   7. Simplified 58.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky} \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin ky \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)\right)}} \cdot \sin ky \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}\right)}} \cdot \sin ky \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)\right)}}} \cdot \sin ky \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \color{blue}{\cos \left(-2 \cdot kx\right)}\right)}} \cdot \sin ky \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \color{blue}{\cos \left(-2 \cdot kx\right)}\right)}} \cdot \sin ky \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \color{blue}{\left(kx \cdot -2\right)}\right)}} \cdot \sin ky \]

      8. lower-*.f64 66.6

         \[\leadsto \frac{\sin th}{\sqrt{0.5 \cdot \left(1 - \cos \color{blue}{\left(kx \cdot -2\right)}\right)}} \cdot \sin ky \]

   7. Simplified 66.6%

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}} \cdot \sin ky \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 35.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;kx \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 5e-201)
   (* (sin th) (/ (sin ky) (sqrt (fma (* 2.0 (* ky ky)) 0.5 (* kx kx)))))
   (if (<= kx 0.86)
     (*
      (fma th (* -0.16666666666666666 (* th th)) th)
      (/
       (sin ky)
       (sqrt
        (fma
         (- 1.0 (cos (+ ky ky)))
         0.5
         (*
          (* kx kx)
          (fma
           (* kx kx)
           (fma
            (* kx kx)
            (fma (* kx kx) -0.0031746031746031746 0.044444444444444446)
            -0.3333333333333333)
           1.0))))))
     (* (* ky (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 5e-201) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((2.0 * (ky * ky)), 0.5, (kx * kx))));
	} else if (kx <= 0.86) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, ((kx * kx) * fma((kx * kx), fma((kx * kx), fma((kx * kx), -0.0031746031746031746, 0.044444444444444446), -0.3333333333333333), 1.0)))));
	} else {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 5e-201)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(2.0 * Float64(ky * ky)), 0.5, Float64(kx * kx)))));
	elseif (kx <= 0.86)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(kx * kx) * fma(Float64(kx * kx), fma(Float64(kx * kx), fma(Float64(kx * kx), -0.0031746031746031746, 0.044444444444444446), -0.3333333333333333), 1.0))))));
	else
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 5e-201], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 0.86], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if kx < 4.9999999999999999e-201

   1. Initial program 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 83.4

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 75.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 50.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   7. Simplified 50.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      3. lower-*.f64 27.7

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   10. Simplified 27.7%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot \left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   if 4.9999999999999999e-201 < kx < 0.859999999999999987

   1. Initial program 90.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 85.8

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 54.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 32.5

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   7. Simplified 32.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\left({kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right) + 1\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\mathsf{fma}\left({kx}^{2}, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}, 1\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) + \color{blue}{\frac{-1}{3}}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}, \frac{-1}{3}\right)}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      13. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{-1}{315} \cdot {kx}^{2} + \frac{2}{45}}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      14. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{-1}{315}} + \frac{2}{45}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{-1}{315}, \frac{2}{45}\right)}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      16. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{-1}{315}, \frac{2}{45}\right), \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      17. lower-*.f64 51.3

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   10. Simplified 51.3%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)}\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 99.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. 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. metadata-eval N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      10. cos-neg N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      13. lower-*.f64 59.6

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;kx \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 35.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;kx \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, 0.044444444444444446, -0.3333333333333333\right), 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 5e-201)
   (* (sin th) (/ (sin ky) (sqrt (fma (* 2.0 (* ky ky)) 0.5 (* kx kx)))))
   (if (<= kx 0.86)
     (*
      (fma th (* -0.16666666666666666 (* th th)) th)
      (/
       (sin ky)
       (sqrt
        (fma
         (- 1.0 (cos (+ ky ky)))
         0.5
         (*
          (* kx kx)
          (fma
           (* kx kx)
           (fma (* kx kx) 0.044444444444444446 -0.3333333333333333)
           1.0))))))
     (* (* ky (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 5e-201) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((2.0 * (ky * ky)), 0.5, (kx * kx))));
	} else if (kx <= 0.86) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, ((kx * kx) * fma((kx * kx), fma((kx * kx), 0.044444444444444446, -0.3333333333333333), 1.0)))));
	} else {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 5e-201)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(2.0 * Float64(ky * ky)), 0.5, Float64(kx * kx)))));
	elseif (kx <= 0.86)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(kx * kx) * fma(Float64(kx * kx), fma(Float64(kx * kx), 0.044444444444444446, -0.3333333333333333), 1.0))))));
	else
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 5e-201], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 0.86], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * 0.044444444444444446 + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if kx < 4.9999999999999999e-201

   1. Initial program 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 83.4

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 75.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 50.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   7. Simplified 50.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      3. lower-*.f64 27.7

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   10. Simplified 27.7%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot \left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   if 4.9999999999999999e-201 < kx < 0.859999999999999987

   1. Initial program 90.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 85.8

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 54.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 32.5

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   7. Simplified 32.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\left({kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) + 1\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, 1\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{2}{45} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{2}{45}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, {kx}^{2} \cdot \frac{2}{45} + \color{blue}{\frac{-1}{3}}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{2}{45}, \frac{-1}{3}\right)}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      13. lower-*.f64 51.3

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 0.044444444444444446, -0.3333333333333333\right), 1\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   10. Simplified 51.3%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, 0.044444444444444446, -0.3333333333333333\right), 1\right)}\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 99.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. 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. metadata-eval N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      10. cos-neg N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      13. lower-*.f64 59.6

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;kx \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, 0.044444444444444446, -0.3333333333333333\right), 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 35.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;kx \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 5e-201)
   (* (sin th) (/ (sin ky) (sqrt (fma (* 2.0 (* ky ky)) 0.5 (* kx kx)))))
   (if (<= kx 0.86)
     (*
      (fma th (* -0.16666666666666666 (* th th)) th)
      (/
       (sin ky)
       (sqrt
        (fma
         (- 1.0 (cos (+ ky ky)))
         0.5
         (* (* kx kx) (fma (* kx kx) -0.3333333333333333 1.0))))))
     (* (* ky (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 5e-201) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((2.0 * (ky * ky)), 0.5, (kx * kx))));
	} else if (kx <= 0.86) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, ((kx * kx) * fma((kx * kx), -0.3333333333333333, 1.0)))));
	} else {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 5e-201)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(2.0 * Float64(ky * ky)), 0.5, Float64(kx * kx)))));
	elseif (kx <= 0.86)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(kx * kx) * fma(Float64(kx * kx), -0.3333333333333333, 1.0))))));
	else
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 5e-201], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 0.86], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if kx < 4.9999999999999999e-201

   1. Initial program 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 83.4

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 75.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 50.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   7. Simplified 50.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      3. lower-*.f64 27.7

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   10. Simplified 27.7%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot \left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   if 4.9999999999999999e-201 < kx < 0.859999999999999987

   1. Initial program 90.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 85.8

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 54.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \frac{1}{2} + \frac{-1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 32.5

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   7. Simplified 32.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {kx}^{2} + 1\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \left(\color{blue}{{kx}^{2} \cdot \frac{-1}{3}} + 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{-1}{3}, 1\right)}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

      8. lower-*.f64 51.3

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   10. Simplified 51.3%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)}\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 99.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. 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. metadata-eval N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      10. cos-neg N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      13. lower-*.f64 59.6

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;kx \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 35.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;kx \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 5e-201)
   (* (sin th) (/ (sin ky) (sqrt (fma (* 2.0 (* ky ky)) 0.5 (* kx kx)))))
   (if (<= kx 0.86)
     (*
      (fma th (* -0.16666666666666666 (* th th)) th)
      (/ (sin ky) (sqrt (fma (- 1.0 (cos (+ ky ky))) 0.5 (* kx kx)))))
     (* (* ky (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 5e-201) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((2.0 * (ky * ky)), 0.5, (kx * kx))));
	} else if (kx <= 0.86) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, (kx * kx))));
	} else {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 5e-201)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(2.0 * Float64(ky * ky)), 0.5, Float64(kx * kx)))));
	elseif (kx <= 0.86)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(kx * kx)))));
	else
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 5e-201], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 0.86], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if kx < 4.9999999999999999e-201

   1. Initial program 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 83.4

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 75.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 50.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   7. Simplified 50.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      3. lower-*.f64 27.7

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   10. Simplified 27.7%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot \left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   if 4.9999999999999999e-201 < kx < 0.859999999999999987

   1. Initial program 90.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 85.8

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 54.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 85.6

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   7. Simplified 85.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, kx \cdot kx\right)}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, kx \cdot kx\right)}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, kx \cdot kx\right)}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, kx \cdot kx\right)}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, kx \cdot kx\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, kx \cdot kx\right)}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, kx \cdot kx\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 51.3

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, kx \cdot kx\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   10. Simplified 51.3%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, kx \cdot kx\right)}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 99.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. 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. metadata-eval N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      10. cos-neg N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      13. lower-*.f64 59.6

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;kx \leq 0.86:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 37.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.86)
   (* (sin th) (/ (sin ky) (sqrt (fma (* 2.0 (* ky ky)) 0.5 (* kx kx)))))
   (* (* ky (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.86) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((2.0 * (ky * ky)), 0.5, (kx * kx))));
	} else {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.86)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(2.0 * Float64(ky * ky)), 0.5, Float64(kx * kx)))));
	else
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.86], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.859999999999999987

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 83.9

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 70.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 58.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   7. Simplified 58.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot {ky}^{2}}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \cdot \sin th \]

      3. lower-*.f64 30.8

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \color{blue}{\left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   10. Simplified 30.8%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{2 \cdot \left(ky \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \cdot \sin th \]

   if 0.859999999999999987 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 99.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. 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. metadata-eval N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      10. cos-neg N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      13. lower-*.f64 59.6

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.86:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(2 \cdot \left(ky \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 25.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.0027:\\ \;\;\;\;\sin th \cdot \frac{1}{kx \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{\left(kx \cdot kx\right) \cdot \sqrt{0.5}}{ky \cdot \sqrt{2}}, \sqrt{0.5} \cdot \frac{\sqrt{2}}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.0027)
   (*
    (sin th)
    (/
     1.0
     (*
      kx
      (fma
       -0.3333333333333333
       (/ (* (* kx kx) (sqrt 0.5)) (* ky (sqrt 2.0)))
       (* (sqrt 0.5) (/ (sqrt 2.0) ky))))))
   (* (* ky (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.0027) {
		tmp = sin(th) * (1.0 / (kx * fma(-0.3333333333333333, (((kx * kx) * sqrt(0.5)) / (ky * sqrt(2.0))), (sqrt(0.5) * (sqrt(2.0) / ky)))));
	} else {
		tmp = (ky * sin(th)) * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.0027)
		tmp = Float64(sin(th) * Float64(1.0 / Float64(kx * fma(-0.3333333333333333, Float64(Float64(Float64(kx * kx) * sqrt(0.5)) / Float64(ky * sqrt(2.0))), Float64(sqrt(0.5) * Float64(sqrt(2.0) / ky))))));
	else
		tmp = Float64(Float64(ky * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.0027], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(kx * N[(-0.3333333333333333 * N[(N[(N[(kx * kx), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.0027000000000000001

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 70.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      7. cos-neg N/A

         \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      12. lower-sqrt.f64 16.9

          \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\color{blue}{\sqrt{0.5}}}{ky}} \cdot \sin th \]

   7. Simplified 16.9%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky}}} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{1}{\color{blue}{kx \cdot \left(\frac{-1}{3} \cdot \frac{{kx}^{2} \cdot \sqrt{\frac{1}{2}}}{ky \cdot \sqrt{2}} + \frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{ky}\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{kx \cdot \left(\frac{-1}{3} \cdot \frac{{kx}^{2} \cdot \sqrt{\frac{1}{2}}}{ky \cdot \sqrt{2}} + \frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{ky}\right)}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{kx \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, \frac{{kx}^{2} \cdot \sqrt{\frac{1}{2}}}{ky \cdot \sqrt{2}}, \frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{ky}\right)}} \cdot \sin th \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\frac{{kx}^{2} \cdot \sqrt{\frac{1}{2}}}{ky \cdot \sqrt{2}}}, \frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{ky}\right)} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \frac{\color{blue}{{kx}^{2} \cdot \sqrt{\frac{1}{2}}}}{ky \cdot \sqrt{2}}, \frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{ky}\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \frac{\color{blue}{\left(kx \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}{ky \cdot \sqrt{2}}, \frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{ky}\right)} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \frac{\color{blue}{\left(kx \cdot kx\right)} \cdot \sqrt{\frac{1}{2}}}{ky \cdot \sqrt{2}}, \frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{ky}\right)} \cdot \sin th \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \frac{\left(kx \cdot kx\right) \cdot \color{blue}{\sqrt{\frac{1}{2}}}}{ky \cdot \sqrt{2}}, \frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{ky}\right)} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \frac{\left(kx \cdot kx\right) \cdot \sqrt{\frac{1}{2}}}{\color{blue}{ky \cdot \sqrt{2}}}, \frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{ky}\right)} \cdot \sin th \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \frac{\left(kx \cdot kx\right) \cdot \sqrt{\frac{1}{2}}}{ky \cdot \color{blue}{\sqrt{2}}}, \frac{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}{ky}\right)} \cdot \sin th \]

      10. associate-/l* N/A

          \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \frac{\left(kx \cdot kx\right) \cdot \sqrt{\frac{1}{2}}}{ky \cdot \sqrt{2}}, \color{blue}{\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{2}}{ky}}\right)} \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \frac{\left(kx \cdot kx\right) \cdot \sqrt{\frac{1}{2}}}{ky \cdot \sqrt{2}}, \color{blue}{\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{2}}{ky}}\right)} \cdot \sin th \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \frac{\left(kx \cdot kx\right) \cdot \sqrt{\frac{1}{2}}}{ky \cdot \sqrt{2}}, \color{blue}{\sqrt{\frac{1}{2}}} \cdot \frac{\sqrt{2}}{ky}\right)} \cdot \sin th \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(\frac{-1}{3}, \frac{\left(kx \cdot kx\right) \cdot \sqrt{\frac{1}{2}}}{ky \cdot \sqrt{2}}, \sqrt{\frac{1}{2}} \cdot \color{blue}{\frac{\sqrt{2}}{ky}}\right)} \cdot \sin th \]

      14. lower-sqrt.f64 20.4

          \[\leadsto \frac{1}{kx \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{\left(kx \cdot kx\right) \cdot \sqrt{0.5}}{ky \cdot \sqrt{2}}, \sqrt{0.5} \cdot \frac{\color{blue}{\sqrt{2}}}{ky}\right)} \cdot \sin th \]

   10. Simplified 20.4%

       \[\leadsto \frac{1}{\color{blue}{kx \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{\left(kx \cdot kx\right) \cdot \sqrt{0.5}}{ky \cdot \sqrt{2}}, \sqrt{0.5} \cdot \frac{\sqrt{2}}{ky}\right)}} \cdot \sin th \]

   if 0.0027000000000000001 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      10. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      11. div-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      19. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      20. lower-+.f64 99.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      21. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, 0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. 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. metadata-eval N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right) + \frac{1}{2}}} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)} + \frac{1}{2}}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right), \frac{1}{2}\right)}}} \]

      10. cos-neg N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot kx\right)}, \frac{1}{2}\right)}} \]

      12. *-commutative N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(kx \cdot -2\right)}, \frac{1}{2}\right)}} \]

      13. lower-*.f64 59.6

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(kx \cdot -2\right)}, 0.5\right)}} \]

   7. Simplified 59.6%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.0027:\\ \;\;\;\;\sin th \cdot \frac{1}{kx \cdot \mathsf{fma}\left(-0.3333333333333333, \frac{\left(kx \cdot kx\right) \cdot \sqrt{0.5}}{ky \cdot \sqrt{2}}, \sqrt{0.5} \cdot \frac{\sqrt{2}}{ky}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 21.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 3.86 \cdot 10^{-91}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \left(\frac{1}{kx} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right)\\ \mathbf{elif}\;ky \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{1}{\sin ky}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 3.86e-91)
   (*
    (sin th)
    (*
     ky
     (*
      (/ 1.0 kx)
      (fma
       ky
       (* ky (fma (* ky ky) 0.008333333333333333 -0.16666666666666666))
       1.0))))
   (if (<= ky 3e-15)
     (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (/ th (sqrt 0.5))))
     (* (sin th) (/ 1.0 (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3.86e-91) {
		tmp = sin(th) * (ky * ((1.0 / kx) * fma(ky, (ky * fma((ky * ky), 0.008333333333333333, -0.16666666666666666)), 1.0)));
	} else if (ky <= 3e-15) {
		tmp = sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * (th / sqrt(0.5)));
	} else {
		tmp = sin(th) * (1.0 / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 3.86e-91)
		tmp = Float64(sin(th) * Float64(ky * Float64(Float64(1.0 / kx) * fma(ky, Float64(ky * fma(Float64(ky * ky), 0.008333333333333333, -0.16666666666666666)), 1.0))));
	elseif (ky <= 3e-15)
		tmp = Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * Float64(th / sqrt(0.5))));
	else
		tmp = Float64(sin(th) * Float64(1.0 / sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 3.86e-91], N[(N[Sin[th], $MachinePrecision] * N[(ky * N[(N[(1.0 / kx), $MachinePrecision] * N[(ky * N[(ky * N[(N[(ky * ky), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 3e-15], N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(th / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 3.86000000000000009e-91

   1. Initial program 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)\right)} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)}\right) \cdot \sin th \]

      3. distribute-lft-in N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right) + {ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)} + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) \cdot \sin th \]

      4. associate-+l+ N/A

         \[\leadsto \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right) + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)}\right) \cdot \sin th \]

      5. associate-*r* N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \frac{-1}{6}\right) \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}} + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2}\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right) \cdot \sin th \]

   5. Simplified 8.4%

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right)} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      2. lower-/.f64 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\color{blue}{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      3. cube-mult N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{\color{blue}{kx \cdot \left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{{kx}^{2}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{\color{blue}{kx \cdot {kx}^{2}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{\left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      7. lower-*.f64 13.2

         \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{\left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

   8. Simplified 13.2%

      \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{kx \cdot \left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

   9. Taylor expanded in kx around 0

      \[\leadsto \left(ky \cdot \left(\color{blue}{\frac{1}{kx}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-/.f64 26.1

          \[\leadsto \left(ky \cdot \left(\color{blue}{\frac{1}{kx}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

   11. Simplified 26.1%

       \[\leadsto \left(ky \cdot \left(\color{blue}{\frac{1}{kx}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

   if 3.86000000000000009e-91 < ky < 3e-15

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      7. cos-neg N/A

         \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

      12. lower-sqrt.f64 75.1

          \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\color{blue}{\sqrt{0.5}}}{ky}} \cdot \sin th \]

   7. Simplified 75.1%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky}}} \cdot \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{ky \cdot th}{\sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \frac{ky \cdot th}{\sqrt{\frac{1}{2}}}} \]

      2. cos-neg N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \frac{ky \cdot th}{\sqrt{\frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot kx\right)}}} \cdot \frac{ky \cdot th}{\sqrt{\frac{1}{2}}} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \left(\color{blue}{2} \cdot kx\right)}} \cdot \frac{ky \cdot th}{\sqrt{\frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \frac{ky \cdot th}{\sqrt{\frac{1}{2}}}} \]

   10. Simplified 28.5%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)} \]

   if 3e-15 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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 99.8

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied egg-rr 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   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. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)}\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(\frac{-1}{6} \cdot {kx}^{2}\right) + kx \cdot 1}\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx \cdot \left(\frac{-1}{6} \cdot {kx}^{2}\right) + \color{blue}{kx}\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(kx, \frac{-1}{6} \cdot {kx}^{2}, kx\right)}\right)} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, \color{blue}{\frac{-1}{6} \cdot {kx}^{2}}, kx\right)\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, \frac{-1}{6} \cdot \color{blue}{\left(kx \cdot kx\right)}, kx\right)\right)} \cdot \sin th \]

      7. lower-*.f64 46.8

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \color{blue}{\left(kx \cdot kx\right)}, kx\right)\right)} \cdot \sin th \]

   7. Simplified 46.8%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)}\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{1}{\sin ky}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sin ky}} \cdot \sin th \]

      2. lower-sin.f64 20.2

         \[\leadsto \frac{1}{\color{blue}{\sin ky}} \cdot \sin th \]

   10. Simplified 20.2%

       \[\leadsto \color{blue}{\frac{1}{\sin ky}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.86 \cdot 10^{-91}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \left(\frac{1}{kx} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right)\\ \mathbf{elif}\;ky \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \frac{th}{\sqrt{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{1}{\sin ky}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 16.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \sin th \cdot \frac{1}{\frac{\sqrt{2} \cdot \left(kx \cdot \sqrt{0.5}\right)}{ky}} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (sin th) (/ 1.0 (/ (* (sqrt 2.0) (* kx (sqrt 0.5))) ky))))

C

double code(double kx, double ky, double th) {
	return sin(th) * (1.0 / ((sqrt(2.0) * (kx * sqrt(0.5))) / ky));
}

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(th) * (1.0d0 / ((sqrt(2.0d0) * (kx * sqrt(0.5d0))) / ky))
end function

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th) * (1.0 / ((Math.sqrt(2.0) * (kx * Math.sqrt(0.5))) / ky));
}

Python

def code(kx, ky, th):
	return math.sin(th) * (1.0 / ((math.sqrt(2.0) * (kx * math.sqrt(0.5))) / ky))

Julia

function code(kx, ky, th)
	return Float64(sin(th) * Float64(1.0 / Float64(Float64(sqrt(2.0) * Float64(kx * sqrt(0.5))) / ky)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th) * (1.0 / ((sqrt(2.0) * (kx * sqrt(0.5))) / ky));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] * N[(1.0 / N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(kx * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

4. Applied egg-rr 78.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}} \cdot \sin th \]

5. Taylor expanded in ky around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   6. lower--.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   7. cos-neg N/A

      \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

   12. lower-sqrt.f64 28.6

       \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\color{blue}{\sqrt{0.5}}}{ky}} \cdot \sin th \]

7. Simplified 28.6%

   \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky}}} \cdot \sin th \]

8. Taylor expanded in kx around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{kx \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{ky}}} \cdot \sin th \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{kx \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{ky}}} \cdot \sin th \]

   2. associate-*r* N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\left(kx \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{2}}}{ky}} \cdot \sin th \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\left(kx \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{2}}}{ky}} \cdot \sin th \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\left(kx \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{2}}{ky}} \cdot \sin th \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{\frac{\left(kx \cdot \color{blue}{\sqrt{\frac{1}{2}}}\right) \cdot \sqrt{2}}{ky}} \cdot \sin th \]

   6. lower-sqrt.f64 20.5

      \[\leadsto \frac{1}{\frac{\left(kx \cdot \sqrt{0.5}\right) \cdot \color{blue}{\sqrt{2}}}{ky}} \cdot \sin th \]

10. Simplified 20.5%

    \[\leadsto \frac{1}{\color{blue}{\frac{\left(kx \cdot \sqrt{0.5}\right) \cdot \sqrt{2}}{ky}}} \cdot \sin th \]

11. Final simplification 20.5%

    \[\leadsto \sin th \cdot \frac{1}{\frac{\sqrt{2} \cdot \left(kx \cdot \sqrt{0.5}\right)}{ky}} \]
12. Add Preprocessing

Alternative 31: 16.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \sin th \cdot \left(ky \cdot \left(\frac{1}{kx} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (sin th)
  (*
   ky
   (*
    (/ 1.0 kx)
    (fma
     ky
     (* ky (fma (* ky ky) 0.008333333333333333 -0.16666666666666666))
     1.0)))))

C

double code(double kx, double ky, double th) {
	return sin(th) * (ky * ((1.0 / kx) * fma(ky, (ky * fma((ky * ky), 0.008333333333333333, -0.16666666666666666)), 1.0)));
}

Julia

function code(kx, ky, th)
	return Float64(sin(th) * Float64(ky * Float64(Float64(1.0 / kx) * fma(ky, Float64(ky * fma(Float64(ky * ky), 0.008333333333333333, -0.16666666666666666)), 1.0))))
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] * N[(ky * N[(N[(1.0 / kx), $MachinePrecision] * N[(ky * N[(ky * N[(N[(ky * ky), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in ky around 0

   \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)\right)} \cdot \sin th \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)\right)} \cdot \sin th \]

   2. +-commutative N/A

      \[\leadsto \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)}\right) \cdot \sin th \]

   3. distribute-lft-in N/A

      \[\leadsto \left(ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right) + {ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right)} + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) \cdot \sin th \]

   4. associate-+l+ N/A

      \[\leadsto \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right) + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)}\right) \cdot \sin th \]

   5. associate-*r* N/A

      \[\leadsto \left(ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \frac{-1}{6}\right) \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}} + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right) \cdot \sin th \]

   6. *-commutative N/A

      \[\leadsto \left(ky \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2}\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}} + \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot \left({ky}^{2} \cdot \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right) + \sqrt{\frac{1}{{\sin kx}^{5}}}\right)\right)\right) \cdot \sin th \]

5. Simplified 6.8%

   \[\leadsto \color{blue}{\left(ky \cdot \left(\sqrt{\frac{1}{{\sin kx}^{5}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right)} \cdot \sin th \]

6. Taylor expanded in kx around 0

   \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]
7. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

   2. lower-/.f64 N/A

      \[\leadsto \left(ky \cdot \left(\sqrt{\color{blue}{\frac{1}{{kx}^{3}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

   3. cube-mult N/A

      \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{\color{blue}{kx \cdot \left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

   4. unpow2 N/A

      \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{{kx}^{2}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

   5. lower-*.f64 N/A

      \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{\color{blue}{kx \cdot {kx}^{2}}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

   6. unpow2 N/A

      \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{\left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

   7. lower-*.f64 10.2

      \[\leadsto \left(ky \cdot \left(\sqrt{\frac{1}{kx \cdot \color{blue}{\left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

8. Simplified 10.2%

   \[\leadsto \left(ky \cdot \left(\color{blue}{\sqrt{\frac{1}{kx \cdot \left(kx \cdot kx\right)}}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

9. Taylor expanded in kx around 0

   \[\leadsto \left(ky \cdot \left(\color{blue}{\frac{1}{kx}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]
10. Step-by-step derivation

    1. lower-/.f64 19.9

       \[\leadsto \left(ky \cdot \left(\color{blue}{\frac{1}{kx}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

11. Simplified 19.9%

    \[\leadsto \left(ky \cdot \left(\color{blue}{\frac{1}{kx}} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

12. Final simplification 19.9%

    \[\leadsto \sin th \cdot \left(ky \cdot \left(\frac{1}{kx} \cdot \mathsf{fma}\left(ky, ky \cdot \mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\right) \]
13. Add Preprocessing

Alternative 32: 16.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \frac{ky \cdot \sin th}{kx} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (/ (* ky (sin th)) kx))

C

double code(double kx, double ky, double th) {
	return (ky * sin(th)) / kx;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (ky * sin(th)) / kx
end function

Java

public static double code(double kx, double ky, double th) {
	return (ky * Math.sin(th)) / kx;
}

Python

def code(kx, ky, th):
	return (ky * math.sin(th)) / kx

Julia

function code(kx, ky, th)
	return Float64(Float64(ky * sin(th)) / kx)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (ky * sin(th)) / kx;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision]

Derivation

1. Initial program 93.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

4. Applied egg-rr 78.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}} \cdot \sin th \]

5. Taylor expanded in ky around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   6. lower--.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   7. cos-neg N/A

      \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

   12. lower-sqrt.f64 28.6

       \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\color{blue}{\sqrt{0.5}}}{ky}} \cdot \sin th \]

7. Simplified 28.6%

   \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky}}} \cdot \sin th \]

8. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{kx} \]

   3. lower-sin.f64 19.5

      \[\leadsto \frac{ky \cdot \color{blue}{\sin th}}{kx} \]

10. Simplified 19.5%

    \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]
11. Add Preprocessing

Alternative 33: 8.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \sin th \cdot \frac{1}{kx} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (sin th) (/ 1.0 kx)))

C

double code(double kx, double ky, double th) {
	return sin(th) * (1.0 / kx);
}

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(th) * (1.0d0 / kx)
end function

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th) * (1.0 / kx);
}

Python

def code(kx, ky, th):
	return math.sin(th) * (1.0 / kx)

Julia

function code(kx, ky, th)
	return Float64(sin(th) * Float64(1.0 / kx))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th) * (1.0 / kx);
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] * N[(1.0 / kx), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

4. Applied egg-rr 78.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}} \cdot \sin th \]

5. Taylor expanded in ky around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   6. lower--.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   7. cos-neg N/A

      \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{\sqrt{1 - \cos \color{blue}{\left(kx \cdot -2\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{ky}} \cdot \sin th \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \color{blue}{\frac{\sqrt{\frac{1}{2}}}{ky}}} \cdot \sin th \]

   12. lower-sqrt.f64 28.6

       \[\leadsto \frac{1}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\color{blue}{\sqrt{0.5}}}{ky}} \cdot \sin th \]

7. Simplified 28.6%

   \[\leadsto \frac{1}{\color{blue}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \frac{\sqrt{0.5}}{ky}}} \cdot \sin th \]

8. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\frac{1}{kx}} \cdot \sin th \]
9. Step-by-step derivation

   1. lower-/.f64 10.4

      \[\leadsto \color{blue}{\frac{1}{kx}} \cdot \sin th \]

10. Simplified 10.4%

    \[\leadsto \color{blue}{\frac{1}{kx}} \cdot \sin th \]

11. Final simplification 10.4%

    \[\leadsto \sin th \cdot \frac{1}{kx} \]
12. Add Preprocessing

Alternative 34: 3.4% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{th}{ky \cdot ky} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (/ th (* ky ky)))

C

double code(double kx, double ky, double th) {
	return th / (ky * ky);
}

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 / (ky * ky)
end function

Java

public static double code(double kx, double ky, double th) {
	return th / (ky * ky);
}

Python

def code(kx, ky, th):
	return th / (ky * ky)

Julia

function code(kx, ky, th)
	return Float64(th / Float64(ky * ky))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th / (ky * ky);
end

Wolfram

code[kx_, ky_, th_] := N[(th / N[(ky * ky), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\frac{\sin th}{{\sin ky}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin th}{{\sin ky}^{3}}} \]

   2. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin th}}{{\sin ky}^{3}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{\sin th}{\color{blue}{{\sin ky}^{3}}} \]

   4. lower-sin.f64 10.7

      \[\leadsto \frac{\sin th}{{\color{blue}{\sin ky}}^{3}} \]

5. Simplified 10.7%

   \[\leadsto \color{blue}{\frac{\sin th}{{\sin ky}^{3}}} \]

6. Taylor expanded in ky around 0

   \[\leadsto \frac{\sin th}{\color{blue}{{ky}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\sin th}{\color{blue}{ky \cdot ky}} \]

   2. lower-*.f64 3.4

      \[\leadsto \frac{\sin th}{\color{blue}{ky \cdot ky}} \]

8. Simplified 3.4%

   \[\leadsto \frac{\sin th}{\color{blue}{ky \cdot ky}} \]

9. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{th}{{ky}^{3}}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{th}{{ky}^{3}}} \]

    2. cube-mult N/A

       \[\leadsto \frac{th}{\color{blue}{ky \cdot \left(ky \cdot ky\right)}} \]

    3. unpow2 N/A

       \[\leadsto \frac{th}{ky \cdot \color{blue}{{ky}^{2}}} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{th}{\color{blue}{ky \cdot {ky}^{2}}} \]

    5. unpow2 N/A

       \[\leadsto \frac{th}{ky \cdot \color{blue}{\left(ky \cdot ky\right)}} \]

    6. lower-*.f64 3.4

       \[\leadsto \frac{th}{ky \cdot \color{blue}{\left(ky \cdot ky\right)}} \]

11. Simplified 3.4%

    \[\leadsto \color{blue}{\frac{th}{ky \cdot \left(ky \cdot ky\right)}} \]

12. Taylor expanded in th around 0

    \[\leadsto \color{blue}{\frac{th}{{ky}^{2}}} \]
13. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{th}{{ky}^{2}}} \]

    2. unpow2 N/A

       \[\leadsto \frac{th}{\color{blue}{ky \cdot ky}} \]

    3. lower-*.f64 3.4

       \[\leadsto \frac{th}{\color{blue}{ky \cdot ky}} \]

14. Simplified 3.4%

    \[\leadsto \color{blue}{\frac{th}{ky \cdot ky}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))