Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 13.2s

Alternatives: 28

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 67.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 6 regimes

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

   1. Initial program 85.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 61.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 61.2

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

   7. Applied rewrites 61.2%

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

   9. Applied rewrites 61.5%

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

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

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 51.9%

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

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

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sin-mult N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2 N/A

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

      10. cos-diff N/A

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

      11. cos-sin-sum N/A

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

      12. lower--.f64 N/A

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

      13. count-2 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-+.f64 76.7

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

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 73.0

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

   7. Applied rewrites 73.0%

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

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

   1. Initial program 98.9%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 52.6

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

   5. Applied rewrites 52.6%

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

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

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 51.2

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

   7. Applied rewrites 51.2%

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

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

   1. Initial program 87.7%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-sin.f64 91.1

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 90.9%

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

4. Final simplification 67.5%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))