Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.9% → 99.7%

Time: 13.6s

Alternatives: 32

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% 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 96.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \cos \left(kx + kx\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_4 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(t\_1, 0.5, t\_3\right)}}\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_2 + ky \cdot ky}}\\ \mathbf{elif}\;t\_4 \leq 0.99:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t\_1, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right) \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(-0.0001984126984126984, th \cdot th, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- 1.0 (cos (+ kx kx))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_2 t_3)))))
   (if (<= t_4 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_4 -0.001)
       (/ (* (sin ky) th) (sqrt (fma t_1 0.5 t_3)))
       (if (<= t_4 0.05)
         (* (sin th) (/ (sin ky) (sqrt (+ t_2 (* ky ky)))))
         (if (<= t_4 0.99)
           (*
            (*
             (sin ky)
             (sqrt (/ 1.0 (fma t_1 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky))))))))
            (*
             th
             (fma
              (* th th)
              (fma
               (* th th)
               (fma -0.0001984126984126984 (* th th) 0.008333333333333333)
               -0.16666666666666666)
              1.0)))
           (*
            (sin th)
            (/
             (sin ky)
             (hypot
              (sin ky)
              (fma kx (* -0.16666666666666666 (* kx kx)) kx))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = 1.0 - cos((kx + kx));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_4 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(fma(t_1, 0.5, t_3));
	} else if (t_4 <= 0.05) {
		tmp = sin(th) * (sin(ky) / sqrt((t_2 + (ky * ky))));
	} else if (t_4 <= 0.99) {
		tmp = (sin(ky) * sqrt((1.0 / fma(t_1, 0.5, (0.5 + (-0.5 * cos((ky + ky)))))))) * (th * fma((th * th), fma((th * th), fma(-0.0001984126984126984, (th * th), 0.008333333333333333), -0.16666666666666666), 1.0));
	} else {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(1.0 - cos(Float64(kx + kx)))
	t_2 = sin(kx) ^ 2.0
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_4 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(fma(t_1, 0.5, t_3)));
	elseif (t_4 <= 0.05)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(t_2 + Float64(ky * ky)))));
	elseif (t_4 <= 0.99)
		tmp = Float64(Float64(sin(ky) * sqrt(Float64(1.0 / fma(t_1, 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))))) * Float64(th * fma(Float64(th * th), fma(Float64(th * th), fma(-0.0001984126984126984, Float64(th * th), 0.008333333333333333), -0.16666666666666666), 1.0)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(t$95$1 * 0.5 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.99], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(t$95$1 * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * N[(-0.0001984126984126984 * N[(th * th), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. sqr-sin-a N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 45.9

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

   9. Applied rewrites 45.9%

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

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.98999999999999999

   1. Initial program 99.4%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 55.2

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

   7. Applied rewrites 55.2%

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

   if 0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 95.4%

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

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

   4. Applied rewrites 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. Applied rewrites 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, {\sin ky}^{2}\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.99:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right) \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(-0.0001984126984126984, th \cdot th, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_3 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{t\_1}}\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_2 + ky \cdot ky}}\\ \mathbf{elif}\;t\_3 \leq 0.99:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{t\_1}}\right) \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(-0.0001984126984126984, th \cdot th, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky))))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_3 -0.001)
       (/ (* (sin ky) th) (sqrt t_1))
       (if (<= t_3 0.05)
         (* (sin th) (/ (sin ky) (sqrt (+ t_2 (* ky ky)))))
         (if (<= t_3 0.99)
           (*
            (* (sin ky) (sqrt (/ 1.0 t_1)))
            (*
             th
             (fma
              (* th th)
              (fma
               (* th th)
               (fma -0.0001984126984126984 (* th th) 0.008333333333333333)
               -0.16666666666666666)
              1.0)))
           (*
            (sin th)
            (/
             (sin ky)
             (hypot
              (sin ky)
              (fma kx (* -0.16666666666666666 (* kx kx)) kx))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_3 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(t_1);
	} else if (t_3 <= 0.05) {
		tmp = sin(th) * (sin(ky) / sqrt((t_2 + (ky * ky))));
	} else if (t_3 <= 0.99) {
		tmp = (sin(ky) * sqrt((1.0 / t_1))) * (th * fma((th * th), fma((th * th), fma(-0.0001984126984126984, (th * th), 0.008333333333333333), -0.16666666666666666), 1.0));
	} else {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_3 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(t_1));
	elseif (t_3 <= 0.05)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(t_2 + Float64(ky * ky)))));
	elseif (t_3 <= 0.99)
		tmp = Float64(Float64(sin(ky) * sqrt(Float64(1.0 / t_1))) * Float64(th * fma(Float64(th * th), fma(Float64(th * th), fma(-0.0001984126984126984, Float64(th * th), 0.008333333333333333), -0.16666666666666666), 1.0)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.99], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * N[(-0.0001984126984126984 * N[(th * th), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.98999999999999999

   1. Initial program 99.4%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 55.2

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

   7. Applied rewrites 55.2%

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

   if 0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 95.4%

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

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

   4. Applied rewrites 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. Applied rewrites 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.99:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right) \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(-0.0001984126984126984, th \cdot th, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_3 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{t\_1}}\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{t\_2 + ky \cdot ky}}\\ \mathbf{elif}\;t\_3 \leq 0.99:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{t\_1}}\right) \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(-0.0001984126984126984, th \cdot th, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky))))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_3 -0.001)
       (/ (* (sin ky) th) (sqrt t_1))
       (if (<= t_3 0.05)
         (*
          (sin th)
          (/
           (fma ky (* -0.16666666666666666 (* ky ky)) ky)
           (sqrt (+ t_2 (* ky ky)))))
         (if (<= t_3 0.99)
           (*
            (* (sin ky) (sqrt (/ 1.0 t_1)))
            (*
             th
             (fma
              (* th th)
              (fma
               (* th th)
               (fma -0.0001984126984126984 (* th th) 0.008333333333333333)
               -0.16666666666666666)
              1.0)))
           (*
            (sin th)
            (/
             (sin ky)
             (hypot
              (sin ky)
              (fma kx (* -0.16666666666666666 (* kx kx)) kx))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_3 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(t_1);
	} else if (t_3 <= 0.05) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sqrt((t_2 + (ky * ky))));
	} else if (t_3 <= 0.99) {
		tmp = (sin(ky) * sqrt((1.0 / t_1))) * (th * fma((th * th), fma((th * th), fma(-0.0001984126984126984, (th * th), 0.008333333333333333), -0.16666666666666666), 1.0));
	} else {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_3 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(t_1));
	elseif (t_3 <= 0.05)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sqrt(Float64(t_2 + Float64(ky * ky)))));
	elseif (t_3 <= 0.99)
		tmp = Float64(Float64(sin(ky) * sqrt(Float64(1.0 / t_1))) * Float64(th * fma(Float64(th * th), fma(Float64(th * th), fma(-0.0001984126984126984, Float64(th * th), 0.008333333333333333), -0.16666666666666666), 1.0)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.99], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * N[(-0.0001984126984126984 * N[(th * th), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.98999999999999999

   1. Initial program 99.4%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 55.2

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

   7. Applied rewrites 55.2%

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

   if 0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 95.4%

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

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

   4. Applied rewrites 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. Applied rewrites 100.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.99:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right) \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(-0.0001984126984126984, th \cdot th, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_3 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{t\_1}}\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{t\_2 + ky \cdot ky}}\\ \mathbf{elif}\;t\_3 \leq 0.93:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{t\_1}}\right) \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(-0.0001984126984126984, th \cdot th, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky))))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_3 -0.001)
       (/ (* (sin ky) th) (sqrt t_1))
       (if (<= t_3 0.05)
         (*
          (sin th)
          (/
           (fma ky (* -0.16666666666666666 (* ky ky)) ky)
           (sqrt (+ t_2 (* ky ky)))))
         (if (<= t_3 0.93)
           (*
            (* (sin ky) (sqrt (/ 1.0 t_1)))
            (*
             th
             (fma
              (* th th)
              (fma
               (* th th)
               (fma -0.0001984126984126984 (* th th) 0.008333333333333333)
               -0.16666666666666666)
              1.0)))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_3 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(t_1);
	} else if (t_3 <= 0.05) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sqrt((t_2 + (ky * ky))));
	} else if (t_3 <= 0.93) {
		tmp = (sin(ky) * sqrt((1.0 / t_1))) * (th * fma((th * th), fma((th * th), fma(-0.0001984126984126984, (th * th), 0.008333333333333333), -0.16666666666666666), 1.0));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_3 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(t_1));
	elseif (t_3 <= 0.05)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sqrt(Float64(t_2 + Float64(ky * ky)))));
	elseif (t_3 <= 0.93)
		tmp = Float64(Float64(sin(ky) * sqrt(Float64(1.0 / t_1))) * Float64(th * fma(Float64(th * th), fma(Float64(th * th), fma(-0.0001984126984126984, Float64(th * th), 0.008333333333333333), -0.16666666666666666), 1.0)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.93], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * N[(-0.0001984126984126984 * N[(th * th), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.930000000000000049

   1. Initial program 99.4%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

   if 0.930000000000000049 < (/.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 95.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 96.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.93:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right) \cdot \left(th \cdot \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(th \cdot th, \mathsf{fma}\left(-0.0001984126984126984, th \cdot th, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \cos \left(kx + kx\right)\\ t_2 := \cos \left(ky + ky\right)\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_4 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_4 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(t\_1, 0.5, 0.5 + -0.5 \cdot t\_2\right)}}\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{t\_3 + ky \cdot ky}}\\ \mathbf{elif}\;t\_4 \leq 0.93:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t\_1, 0.5, \mathsf{fma}\left(t\_2, -0.5, 0.5\right)\right)}}\right) \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- 1.0 (cos (+ kx kx))))
        (t_2 (cos (+ ky ky)))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin ky) 2.0))))))
   (if (<= t_4 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_4 -0.001)
       (/ (* (sin ky) th) (sqrt (fma t_1 0.5 (+ 0.5 (* -0.5 t_2)))))
       (if (<= t_4 0.05)
         (*
          (sin th)
          (/
           (fma ky (* -0.16666666666666666 (* ky ky)) ky)
           (sqrt (+ t_3 (* ky ky)))))
         (if (<= t_4 0.93)
           (*
            (* (sin ky) (sqrt (/ 1.0 (fma t_1 0.5 (fma t_2 -0.5 0.5)))))
            (fma
             th
             (*
              (* th th)
              (fma 0.008333333333333333 (* th th) -0.16666666666666666))
             th))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = 1.0 - cos((kx + kx));
	double t_2 = cos((ky + ky));
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_4 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_4 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(fma(t_1, 0.5, (0.5 + (-0.5 * t_2))));
	} else if (t_4 <= 0.05) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sqrt((t_3 + (ky * ky))));
	} else if (t_4 <= 0.93) {
		tmp = (sin(ky) * sqrt((1.0 / fma(t_1, 0.5, fma(t_2, -0.5, 0.5))))) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(1.0 - cos(Float64(kx + kx)))
	t_2 = cos(Float64(ky + ky))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_4 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(fma(t_1, 0.5, Float64(0.5 + Float64(-0.5 * t_2)))));
	elseif (t_4 <= 0.05)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sqrt(Float64(t_3 + Float64(ky * ky)))));
	elseif (t_4 <= 0.93)
		tmp = Float64(Float64(sin(ky) * sqrt(Float64(1.0 / fma(t_1, 0.5, fma(t_2, -0.5, 0.5))))) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(t$95$1 * 0.5 + N[(0.5 + N[(-0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.93], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(t$95$1 * 0.5 + N[(t$95$2 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.930000000000000049

   1. Initial program 99.4%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.2

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

   7. Applied rewrites 56.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{\color{blue}{0}}{ky - ky}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin ky\right) \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\frac{1}{120}, th \cdot th, \frac{-1}{6}\right), th\right) \]

      8. +-inverses N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-inverses N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right) \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin ky\right) \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\frac{1}{120}, th \cdot th, \frac{-1}{6}\right), th\right) \]

      12. +-inverses N/A

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

      13. flip-+ N/A

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

      14. lift-+.f64 N/A

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

      15. lower-fma.f64 14.2

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

   9. Applied rewrites 56.2%

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

   if 0.930000000000000049 < (/.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 95.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 96.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.93:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\right) \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \sin ky \cdot th\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ t_4 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_4}}\\ \mathbf{elif}\;t\_3 \leq -0.001:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{t\_1 + ky \cdot ky}}\\ \mathbf{elif}\;t\_3 \leq 0.9999996186923419:\\ \;\;\;\;t\_2 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (* (sin ky) th))
        (t_3 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0)))))
        (t_4 (fma -0.5 (cos (* ky -2.0)) 0.5)))
   (if (<= t_3 -0.98)
     (* (sin ky) (/ (sin th) (sqrt t_4)))
     (if (<= t_3 -0.001)
       (/
        t_2
        (sqrt
         (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky)))))))
       (if (<= t_3 0.05)
         (*
          (sin th)
          (/
           (fma ky (* -0.16666666666666666 (* ky ky)) ky)
           (sqrt (+ t_1 (* ky ky)))))
         (if (<= t_3 0.9999996186923419)
           (* t_2 (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_4))))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) * th;
	double t_3 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double t_4 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double tmp;
	if (t_3 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(t_4));
	} else if (t_3 <= -0.001) {
		tmp = t_2 / sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))));
	} else if (t_3 <= 0.05) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sqrt((t_1 + (ky * ky))));
	} else if (t_3 <= 0.9999996186923419) {
		tmp = t_2 * sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_4)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) * th)
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	t_4 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_4)));
	elseif (t_3 <= -0.001)
		tmp = Float64(t_2 / sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))));
	elseif (t_3 <= 0.05)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sqrt(Float64(t_1 + Float64(ky * ky)))));
	elseif (t_3 <= 0.9999996186923419)
		tmp = Float64(t_2 * sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_4))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.001], N[(t$95$2 / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9999996186923419], N[(t$95$2 * N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999961869234189

   1. Initial program 99.4%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

   7. Applied rewrites 56.2%

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

   if 0.99999961869234189 < (/.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 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 100.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9999996186923419:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ t_3 := \frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{if}\;t\_2 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_2 \leq -0.001:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{t\_1 + ky \cdot ky}}\\ \mathbf{elif}\;t\_2 \leq 0.9999996186923419:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0)))))
        (t_3
         (/
          (* (sin ky) th)
          (sqrt
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (+ 0.5 (* -0.5 (cos (+ ky ky)))))))))
   (if (<= t_2 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_2 -0.001)
       t_3
       (if (<= t_2 0.05)
         (*
          (sin th)
          (/
           (fma ky (* -0.16666666666666666 (* ky ky)) ky)
           (sqrt (+ t_1 (* ky ky)))))
         (if (<= t_2 0.9999996186923419) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double t_3 = (sin(ky) * th) / sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_2 <= -0.001) {
		tmp = t_3;
	} else if (t_2 <= 0.05) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sqrt((t_1 + (ky * ky))));
	} else if (t_2 <= 0.9999996186923419) {
		tmp = t_3;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	t_3 = Float64(Float64(sin(ky) * th) / sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))))
	tmp = 0.0
	if (t_2 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_2 <= -0.001)
		tmp = t_3;
	elseif (t_2 <= 0.05)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sqrt(Float64(t_1 + Float64(ky * ky)))));
	elseif (t_2 <= 0.9999996186923419)
		tmp = t_3;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.001], t$95$3, If[LessEqual[t$95$2, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999996186923419], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3 or 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999961869234189

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 50.8

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

   7. Applied rewrites 50.8%

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

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 0.99999961869234189 < (/.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 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 100.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9999996186923419:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \cos \left(kx + kx\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(t\_1, 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{if}\;t\_2 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_2 \leq -0.001:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_1, 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right)\\ \mathbf{elif}\;t\_2 \leq 0.9999996186923419:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- 1.0 (cos (+ kx kx))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3
         (/
          (* (sin ky) th)
          (sqrt (fma t_1 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky)))))))))
   (if (<= t_2 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_2 -0.001)
       t_3
       (if (<= t_2 0.05)
         (*
          (sin th)
          (*
           (sqrt
            (/
             1.0
             (fma
              t_1
              0.5
              (* (* ky ky) (fma (* ky ky) -0.3333333333333333 1.0)))))
           (fma
            ky
            (*
             (* ky ky)
             (fma
              (* ky ky)
              (fma -0.0001984126984126984 (* ky ky) 0.008333333333333333)
              -0.16666666666666666))
            ky)))
         (if (<= t_2 0.9999996186923419) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = 1.0 - cos((kx + kx));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = (sin(ky) * th) / sqrt(fma(t_1, 0.5, (0.5 + (-0.5 * cos((ky + ky))))));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_2 <= -0.001) {
		tmp = t_3;
	} else if (t_2 <= 0.05) {
		tmp = sin(th) * (sqrt((1.0 / fma(t_1, 0.5, ((ky * ky) * fma((ky * ky), -0.3333333333333333, 1.0))))) * fma(ky, ((ky * ky) * fma((ky * ky), fma(-0.0001984126984126984, (ky * ky), 0.008333333333333333), -0.16666666666666666)), ky));
	} else if (t_2 <= 0.9999996186923419) {
		tmp = t_3;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(1.0 - cos(Float64(kx + kx)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(Float64(sin(ky) * th) / sqrt(fma(t_1, 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))))
	tmp = 0.0
	if (t_2 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_2 <= -0.001)
		tmp = t_3;
	elseif (t_2 <= 0.05)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(t_1, 0.5, Float64(Float64(ky * ky) * fma(Float64(ky * ky), -0.3333333333333333, 1.0))))) * fma(ky, Float64(Float64(ky * ky) * fma(Float64(ky * ky), fma(-0.0001984126984126984, Float64(ky * ky), 0.008333333333333333), -0.16666666666666666)), ky)));
	elseif (t_2 <= 0.9999996186923419)
		tmp = t_3;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(1.0 - N[Cos[N[(kx + kx), $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[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(t$95$1 * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.001], t$95$3, If[LessEqual[t$95$2, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(t$95$1 * 0.5 + N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(-0.0001984126984126984 * N[(ky * ky), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999996186923419], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3 or 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999961869234189

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.3

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 50.8

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

   7. Applied rewrites 50.8%

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

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) + ky \cdot 1\right)}\right) \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) + \color{blue}{ky}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      8. sub-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, ky\right)\right) \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \color{blue}{\frac{-1}{6}}\right), ky\right)\right) \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      13. +-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-1}{5040} \cdot {ky}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {ky}^{2}, \frac{1}{120}\right)}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      15. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{ky \cdot ky}, \frac{1}{120}\right), \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      16. lower-*.f64 76.0

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, \color{blue}{ky \cdot ky}, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right) \cdot \sin th \]

   10. Applied rewrites 76.0%

       \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)}\right) \cdot \sin th \]

   if 0.99999961869234189 < (/.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 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 100.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9999996186923419:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.708:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.71:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.708)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 0.71)
       (/ (* (sin ky) (sin th)) (sqrt (* 0.5 (- 1.0 (cos (* kx -2.0))))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.708) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 0.71) {
		tmp = (sin(ky) * sin(th)) / sqrt((0.5 * (1.0 - cos((kx * -2.0)))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.708)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 0.71)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(kx * -2.0))))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.708], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.71], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.70799999999999996

   1. Initial program 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 87.0

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 55.7

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

   7. Applied rewrites 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.9

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

   9. Applied rewrites 55.9%

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

   if -0.70799999999999996 < (/.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))))) < 0.70999999999999996

   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-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 97.5

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower--.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 58.5

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

   7. Applied rewrites 58.5%

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

   if 0.70999999999999996 < (/.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 96.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 82.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.708:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.71:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(kx \cdot -2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.708:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.71:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.708)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 0.71)
       (* (sin th) (* (sin ky) (sqrt (/ 2.0 (- 1.0 (cos (* kx -2.0)))))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.708) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 0.71) {
		tmp = sin(th) * (sin(ky) * sqrt((2.0 / (1.0 - cos((kx * -2.0))))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.708)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 0.71)
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.708], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.71], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.70799999999999996

   1. Initial program 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 87.0

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sin-mult N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 55.7

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

   7. Applied rewrites 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.9

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

   9. Applied rewrites 55.9%

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

   if -0.70799999999999996 < (/.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))))) < 0.70999999999999996

   1. Initial program 99.4%

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower--.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 58.5

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

   7. Applied rewrites 58.5%

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

   if 0.70999999999999996 < (/.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 96.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 82.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.708:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.71:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.16:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.001)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 0.16)
       (*
        (sin th)
        (*
         (sqrt
          (/
           1.0
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (* (* ky ky) (fma (* ky ky) -0.3333333333333333 1.0)))))
         (fma
          ky
          (*
           (* ky ky)
           (fma
            (* ky ky)
            (fma -0.0001984126984126984 (* ky ky) 0.008333333333333333)
            -0.16666666666666666))
          ky)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.001) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 0.16) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, ((ky * ky) * fma((ky * ky), -0.3333333333333333, 1.0))))) * fma(ky, ((ky * ky) * fma((ky * ky), fma(-0.0001984126984126984, (ky * ky), 0.008333333333333333), -0.16666666666666666)), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 0.16)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(Float64(ky * ky) * fma(Float64(ky * ky), -0.3333333333333333, 1.0))))) * fma(ky, Float64(Float64(ky * ky) * fma(Float64(ky * ky), fma(-0.0001984126984126984, Float64(ky * ky), 0.008333333333333333), -0.16666666666666666)), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.001], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.16], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(-0.0001984126984126984 * N[(ky * ky), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 89.7

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}} + {\sin ky}^{2}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin ky}^{2}}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}}} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)} + \frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. lower-*.f64 47.6

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \]

   7. Applied rewrites 47.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \cdot \sin ky} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \cdot \sin ky} \]

      6. lower-/.f64 47.7

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \cdot \sin ky \]

   9. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \cdot \sin ky} \]

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.160000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {ky}^{2} + 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \left(\color{blue}{{ky}^{2} \cdot \frac{-1}{3}} + 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{3}, 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{3}, 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-*.f64 74.4

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.3333333333333333, 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Applied rewrites 74.4%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) + ky \cdot 1\right)}\right) \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) + \color{blue}{ky}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      8. sub-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, ky\right)\right) \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \color{blue}{\frac{-1}{6}}\right), ky\right)\right) \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      13. +-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-1}{5040} \cdot {ky}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {ky}^{2}, \frac{1}{120}\right)}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      15. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{ky \cdot ky}, \frac{1}{120}\right), \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      16. lower-*.f64 74.2

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, \color{blue}{ky \cdot ky}, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right) \cdot \sin th \]

   10. Applied rewrites 74.2%

       \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)}\right) \cdot \sin th \]

   if 0.160000000000000003 < (/.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 97.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 68.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.16:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.16:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.001)
     (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 0.16)
       (*
        (sin th)
        (*
         (sqrt
          (/
           1.0
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (* (* ky ky) (fma (* ky ky) -0.3333333333333333 1.0)))))
         (fma
          ky
          (*
           (* ky ky)
           (fma
            (* ky ky)
            (fma -0.0001984126984126984 (* ky ky) 0.008333333333333333)
            -0.16666666666666666))
          ky)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.001) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 0.16) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, ((ky * ky) * fma((ky * ky), -0.3333333333333333, 1.0))))) * fma(ky, ((ky * ky) * fma((ky * ky), fma(-0.0001984126984126984, (ky * ky), 0.008333333333333333), -0.16666666666666666)), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 0.16)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(Float64(ky * ky) * fma(Float64(ky * ky), -0.3333333333333333, 1.0))))) * fma(ky, Float64(Float64(ky * ky) * fma(Float64(ky * ky), fma(-0.0001984126984126984, Float64(ky * ky), 0.008333333333333333), -0.16666666666666666)), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.001], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.16], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(-0.0001984126984126984 * N[(ky * ky), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 89.7

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}} + {\sin ky}^{2}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin ky}^{2}}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}}} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)} + \frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. lower-*.f64 47.6

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \]

   7. Applied rewrites 47.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}}} \]

      6. lower-/.f64 47.7

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]

   9. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.160000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {ky}^{2} + 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \left(\color{blue}{{ky}^{2} \cdot \frac{-1}{3}} + 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{3}, 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{3}, 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-*.f64 74.4

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.3333333333333333, 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Applied rewrites 74.4%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) + ky \cdot 1\right)}\right) \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) + \color{blue}{ky}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      8. sub-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, ky\right)\right) \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \color{blue}{\frac{-1}{6}}\right), ky\right)\right) \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      13. +-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-1}{5040} \cdot {ky}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {ky}^{2}, \frac{1}{120}\right)}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      15. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{ky \cdot ky}, \frac{1}{120}\right), \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      16. lower-*.f64 74.2

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, \color{blue}{ky \cdot ky}, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right) \cdot \sin th \]

   10. Applied rewrites 74.2%

       \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)}\right) \cdot \sin th \]

   if 0.160000000000000003 < (/.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 97.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 68.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.16:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.16:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.001)
     (/ (* (sin ky) th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5)))
     (if (<= t_1 0.16)
       (*
        (sin th)
        (*
         (sqrt
          (/
           1.0
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (* (* ky ky) (fma (* ky ky) -0.3333333333333333 1.0)))))
         (fma
          ky
          (*
           (* ky ky)
           (fma
            (* ky ky)
            (fma -0.0001984126984126984 (* ky ky) 0.008333333333333333)
            -0.16666666666666666))
          ky)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5));
	} else if (t_1 <= 0.16) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, ((ky * ky) * fma((ky * ky), -0.3333333333333333, 1.0))))) * fma(ky, ((ky * ky) * fma((ky * ky), fma(-0.0001984126984126984, (ky * ky), 0.008333333333333333), -0.16666666666666666)), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5)));
	elseif (t_1 <= 0.16)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(Float64(ky * ky) * fma(Float64(ky * ky), -0.3333333333333333, 1.0))))) * fma(ky, Float64(Float64(ky * ky) * fma(Float64(ky * ky), fma(-0.0001984126984126984, Float64(ky * ky), 0.008333333333333333), -0.16666666666666666)), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.16], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * N[(-0.0001984126984126984 * N[(ky * ky), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 89.7

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}} + {\sin ky}^{2}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin ky}^{2}}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}}} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)} + \frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. lower-*.f64 47.6

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \]

   7. Applied rewrites 47.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]

      2. lower-sin.f64 26.7

         \[\leadsto \frac{th \cdot \color{blue}{\sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   10. Applied rewrites 26.7%

       \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.160000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {ky}^{2} + 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \left(\color{blue}{{ky}^{2} \cdot \frac{-1}{3}} + 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{3}, 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{3}, 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-*.f64 74.4

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.3333333333333333, 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Applied rewrites 74.4%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)\right)}\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) + ky \cdot 1\right)}\right) \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) + \color{blue}{ky}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      8. sub-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, ky\right)\right) \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) + \color{blue}{\frac{-1}{6}}\right), ky\right)\right) \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      13. +-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{-1}{5040} \cdot {ky}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{5040}, {ky}^{2}, \frac{1}{120}\right)}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      15. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\frac{-1}{5040}, \color{blue}{ky \cdot ky}, \frac{1}{120}\right), \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      16. lower-*.f64 74.2

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, \color{blue}{ky \cdot ky}, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right) \cdot \sin th \]

   10. Applied rewrites 74.2%

       \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)}\right) \cdot \sin th \]

   if 0.160000000000000003 < (/.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 97.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 68.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.16:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(-0.0001984126984126984, ky \cdot ky, 0.008333333333333333\right), -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.16:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.001)
     (/ (* (sin ky) th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5)))
     (if (<= t_1 0.16)
       (*
        (sin th)
        (*
         (sqrt
          (/
           1.0
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (* (* ky ky) (fma (* ky ky) -0.3333333333333333 1.0)))))
         (*
          ky
          (fma
           (* ky ky)
           (fma 0.008333333333333333 (* ky ky) -0.16666666666666666)
           1.0))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5));
	} else if (t_1 <= 0.16) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, ((ky * ky) * fma((ky * ky), -0.3333333333333333, 1.0))))) * (ky * fma((ky * ky), fma(0.008333333333333333, (ky * ky), -0.16666666666666666), 1.0)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5)));
	elseif (t_1 <= 0.16)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(Float64(ky * ky) * fma(Float64(ky * ky), -0.3333333333333333, 1.0))))) * Float64(ky * fma(Float64(ky * ky), fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666), 1.0))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.16], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 89.7

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}} + {\sin ky}^{2}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin ky}^{2}}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}}} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)} + \frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. lower-*.f64 47.6

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \]

   7. Applied rewrites 47.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]

      2. lower-sin.f64 26.7

         \[\leadsto \frac{th \cdot \color{blue}{\sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   10. Applied rewrites 26.7%

       \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.160000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {ky}^{2} + 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \left(\color{blue}{{ky}^{2} \cdot \frac{-1}{3}} + 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{3}, 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{3}, 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-*.f64 74.4

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.3333333333333333, 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Applied rewrites 74.4%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \sin th \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, 1\right)}\right)\right) \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, 1\right)\right)\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, 1\right)\right)\right) \cdot \sin th \]

      6. sub-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right)\right) \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}, 1\right)\right)\right) \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, 1\right)\right)\right) \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), 1\right)\right)\right) \cdot \sin th \]

      10. lower-*.f64 74.2

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(0.008333333333333333, \color{blue}{ky \cdot ky}, -0.16666666666666666\right), 1\right)\right)\right) \cdot \sin th \]

   10. Applied rewrites 74.2%

       \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), 1\right)\right)}\right) \cdot \sin th \]

   if 0.160000000000000003 < (/.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 97.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 68.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.16:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.001)
     (/ (* (sin ky) th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5)))
     (if (<= t_1 0.05)
       (*
        (sin th)
        (*
         (sqrt
          (/
           1.0
           (fma
            (- 1.0 (cos (+ kx kx)))
            0.5
            (* (* ky ky) (fma (* ky ky) -0.3333333333333333 1.0)))))
         (fma ky (* -0.16666666666666666 (* ky ky)) ky)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5));
	} else if (t_1 <= 0.05) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, ((ky * ky) * fma((ky * ky), -0.3333333333333333, 1.0))))) * fma(ky, (-0.16666666666666666 * (ky * ky)), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5)));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(Float64(ky * ky) * fma(Float64(ky * ky), -0.3333333333333333, 1.0))))) * fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(ky * ky), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 89.7

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}} + {\sin ky}^{2}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin ky}^{2}}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}}} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)} + \frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. lower-*.f64 47.6

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \]

   7. Applied rewrites 47.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]

      2. lower-sin.f64 26.7

         \[\leadsto \frac{th \cdot \color{blue}{\sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   10. Applied rewrites 26.7%

       \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {ky}^{2} + 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \left(\color{blue}{{ky}^{2} \cdot \frac{-1}{3}} + 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{3}, 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{3}, 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-*.f64 76.0

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.3333333333333333, 1\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Applied rewrites 76.0%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{\left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)\right)}\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}\right)\right) \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1\right)}\right) \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\frac{-1}{6} \cdot {ky}^{2}}, ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, \frac{-1}{6} \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 75.7

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)\right) \cdot \sin th \]

   10. Applied rewrites 75.7%

       \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}\right) \cdot \sin th \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 97.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right)\right)}} \cdot \mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.001)
     (/ (* (sin ky) th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5)))
     (if (<= t_1 0.05)
       (*
        (sin th)
        (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5));
	} else if (t_1 <= 0.05) {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5)));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 89.7

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}} + {\sin ky}^{2}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin ky}^{2}}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}}} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)} + \frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. lower-*.f64 47.6

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \]

   7. Applied rewrites 47.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]

      2. lower-sin.f64 26.7

         \[\leadsto \frac{th \cdot \color{blue}{\sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   10. Applied rewrites 26.7%

       \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 73.9

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 97.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \sqrt{2}}{\sqrt{1 - \cos \left(kx \cdot -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.001)
     (/ (* (sin ky) th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5)))
     (if (<= t_1 0.05)
       (* (sin th) (/ (* ky (sqrt 2.0)) (sqrt (- 1.0 (cos (* kx -2.0))))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5));
	} else if (t_1 <= 0.05) {
		tmp = sin(th) * ((ky * sqrt(2.0)) / sqrt((1.0 - cos((kx * -2.0)))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5)));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(th) * Float64(Float64(ky * sqrt(2.0)) / sqrt(Float64(1.0 - cos(Float64(kx * -2.0))))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 89.7

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}} + {\sin ky}^{2}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin ky}^{2}}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}}} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)} + \frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. lower-*.f64 47.6

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \]

   7. Applied rewrites 47.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]

      2. lower-sin.f64 26.7

         \[\leadsto \frac{th \cdot \color{blue}{\sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   10. Applied rewrites 26.7%

       \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 73.9

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \]

      3. lower-*.f64 73.9

         \[\leadsto \color{blue}{\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \]

   9. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\sin th \cdot \frac{ky \cdot \sqrt{2}}{\sqrt{1 - \cos \left(kx \cdot -2\right)}}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 97.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \sqrt{2}}{\sqrt{1 - \cos \left(kx \cdot -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 50.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \frac{1}{\sin kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.001)
     (/ (* (sin ky) th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5)))
     (if (<= t_1 2e-8) (* (sin th) (* ky (/ 1.0 (sin kx)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.001) {
		tmp = (sin(ky) * th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5));
	} else if (t_1 <= 2e-8) {
		tmp = sin(th) * (ky * (1.0 / sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.001)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5)));
	elseif (t_1 <= 2e-8)
		tmp = Float64(sin(th) * Float64(ky * Float64(1.0 / sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.001], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-8], N[(N[Sin[th], $MachinePrecision] * N[(ky * N[(1.0 / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 89.7

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \]

      11. sin-mult N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \]

      12. div-inv N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}} + {\sin ky}^{2}}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin ky}^{2}}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}}} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right) + \frac{1}{2}}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)} + \frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right), \frac{1}{2}\right)}}} \]

      5. cos-neg N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \]

      8. lower-*.f64 47.6

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \]

   7. Applied rewrites 47.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \frac{1}{2}\right)}} \]

      2. lower-sin.f64 26.7

         \[\leadsto \frac{th \cdot \color{blue}{\sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   10. Applied rewrites 26.7%

       \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}} \]

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(-1 \cdot \left({ky}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right) + \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \left(-1 \cdot \left({ky}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right) + \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]

      2. mul-1-neg N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left(\mathsf{neg}\left({ky}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right)\right)} + \frac{1}{\sin kx}\right)\right) \cdot \sin th \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{{ky}^{2} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right)\right)} + \frac{1}{\sin kx}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \mathsf{neg}\left(\left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right), \frac{1}{\sin kx}\right)}\right) \cdot \sin th \]

   5. Applied rewrites 48.0%

      \[\leadsto \color{blue}{\left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-0.5}{{\sin kx}^{3}} + \frac{-0.16666666666666666}{\sin kx}, \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{\sin kx}}\right) \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 51.3%

      \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{\sin kx}}\right) \cdot \sin th \]

   if 2e-8 < (/.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 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \frac{1}{\sin kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 30.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 10^{-159}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \frac{-0.5 \cdot \left(ky \cdot ky\right)}{kx \cdot \left(kx \cdot kx\right)}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(ky \cdot \sqrt{2}\right) \cdot \frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 1e-159)
     (* (sin th) (* ky (/ (* -0.5 (* ky ky)) (* kx (* kx kx)))))
     (if (<= t_1 2e-8)
       (*
        (*
         (* ky (sqrt 2.0))
         (/ (fma (* kx kx) (/ 0.08333333333333333 (sqrt 0.5)) (sqrt 0.5)) kx))
        (fma th (* -0.16666666666666666 (* th th)) th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 1e-159) {
		tmp = sin(th) * (ky * ((-0.5 * (ky * ky)) / (kx * (kx * kx))));
	} else if (t_1 <= 2e-8) {
		tmp = ((ky * sqrt(2.0)) * (fma((kx * kx), (0.08333333333333333 / sqrt(0.5)), sqrt(0.5)) / kx)) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 1e-159)
		tmp = Float64(sin(th) * Float64(ky * Float64(Float64(-0.5 * Float64(ky * ky)) / Float64(kx * Float64(kx * kx)))));
	elseif (t_1 <= 2e-8)
		tmp = Float64(Float64(Float64(ky * sqrt(2.0)) * Float64(fma(Float64(kx * kx), Float64(0.08333333333333333 / sqrt(0.5)), sqrt(0.5)) / kx)) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-159], N[(N[Sin[th], $MachinePrecision] * N[(ky * N[(N[(-0.5 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] / N[(kx * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-8], N[(N[(N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(kx * kx), $MachinePrecision] * N[(0.08333333333333333 / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999989e-160

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(-1 \cdot \left({ky}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right) + \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \left(-1 \cdot \left({ky}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right) + \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]

      2. mul-1-neg N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left(\mathsf{neg}\left({ky}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right)\right)} + \frac{1}{\sin kx}\right)\right) \cdot \sin th \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{{ky}^{2} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right)\right)} + \frac{1}{\sin kx}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \mathsf{neg}\left(\left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right), \frac{1}{\sin kx}\right)}\right) \cdot \sin th \]

   5. Applied rewrites 19.2%

      \[\leadsto \color{blue}{\left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-0.5}{{\sin kx}^{3}} + \frac{-0.16666666666666666}{\sin kx}, \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \left(ky \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{ky}^{2}}{{kx}^{3}}}\right)\right) \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 8.4%

      \[\leadsto \left(ky \cdot \frac{\left(ky \cdot ky\right) \cdot -0.5}{\color{blue}{kx \cdot \left(kx \cdot kx\right)}}\right) \cdot \sin th \]

   if 9.99999999999999989e-160 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 67.6

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\frac{\sqrt{\frac{1}{2}} + \frac{1}{12} \cdot \frac{{kx}^{2}}{\sqrt{\frac{1}{2}}}}{kx} \cdot \left(\color{blue}{ky} \cdot \sqrt{2}\right)\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 20.0%

       \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx} \cdot \left(\color{blue}{ky} \cdot \sqrt{2}\right)\right) \cdot \sin th \]

   11. Taylor expanded in th around 0

       \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

       3. *-rgt-identity N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

       6. unpow2 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

       7. lower-*.f64 19.4

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   13. Applied rewrites 19.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   if 2e-8 < (/.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 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-159}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \frac{-0.5 \cdot \left(ky \cdot ky\right)}{kx \cdot \left(kx \cdot kx\right)}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(ky \cdot \sqrt{2}\right) \cdot \frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 30.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-172}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(ky \cdot \sqrt{2}\right) \cdot \frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 5e-172)
     (* -0.16666666666666666 (* th (* th th)))
     (if (<= t_1 2e-8)
       (*
        (*
         (* ky (sqrt 2.0))
         (/ (fma (* kx kx) (/ 0.08333333333333333 (sqrt 0.5)) (sqrt 0.5)) kx))
        (fma th (* -0.16666666666666666 (* th th)) th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-172) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else if (t_1 <= 2e-8) {
		tmp = ((ky * sqrt(2.0)) * (fma((kx * kx), (0.08333333333333333 / sqrt(0.5)), sqrt(0.5)) / kx)) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 5e-172)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	elseif (t_1 <= 2e-8)
		tmp = Float64(Float64(Float64(ky * sqrt(2.0)) * Float64(fma(Float64(kx * kx), Float64(0.08333333333333333 / sqrt(0.5)), sqrt(0.5)) / kx)) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-172], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-8], N[(N[(N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(kx * kx), $MachinePrecision] * N[(0.08333333333333333 / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-172

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.0%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.1%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 10.0%

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   if 4.9999999999999999e-172 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 67.6

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\frac{\sqrt{\frac{1}{2}} + \frac{1}{12} \cdot \frac{{kx}^{2}}{\sqrt{\frac{1}{2}}}}{kx} \cdot \left(\color{blue}{ky} \cdot \sqrt{2}\right)\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 20.0%

       \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx} \cdot \left(\color{blue}{ky} \cdot \sqrt{2}\right)\right) \cdot \sin th \]

   11. Taylor expanded in th around 0

       \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

       3. *-rgt-identity N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

       6. unpow2 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

       7. lower-*.f64 19.4

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   13. Applied rewrites 19.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   if 2e-8 < (/.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 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-172}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(ky \cdot \sqrt{2}\right) \cdot \frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 20.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-172}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(ky \cdot \sqrt{2}\right) \cdot \frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 5e-172)
     (* -0.16666666666666666 (* th (* th th)))
     (if (<= t_1 2e-8)
       (*
        (*
         (* ky (sqrt 2.0))
         (/ (fma (* kx kx) (/ 0.08333333333333333 (sqrt 0.5)) (sqrt 0.5)) kx))
        (fma th (* -0.16666666666666666 (* th th)) th))
       (fma
        th
        (* (* th th) (fma 0.008333333333333333 (* th th) -0.16666666666666666))
        th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-172) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else if (t_1 <= 2e-8) {
		tmp = ((ky * sqrt(2.0)) * (fma((kx * kx), (0.08333333333333333 / sqrt(0.5)), sqrt(0.5)) / kx)) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else {
		tmp = fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 5e-172)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	elseif (t_1 <= 2e-8)
		tmp = Float64(Float64(Float64(ky * sqrt(2.0)) * Float64(fma(Float64(kx * kx), Float64(0.08333333333333333 / sqrt(0.5)), sqrt(0.5)) / kx)) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	else
		tmp = fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-172], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-8], N[(N[(N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(kx * kx), $MachinePrecision] * N[(0.08333333333333333 / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-172

   1. Initial program 95.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.0%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.1%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 10.0%

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   if 4.9999999999999999e-172 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 67.6

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\frac{\sqrt{\frac{1}{2}} + \frac{1}{12} \cdot \frac{{kx}^{2}}{\sqrt{\frac{1}{2}}}}{kx} \cdot \left(\color{blue}{ky} \cdot \sqrt{2}\right)\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 20.0%

       \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx} \cdot \left(\color{blue}{ky} \cdot \sqrt{2}\right)\right) \cdot \sin th \]

   11. Taylor expanded in th around 0

       \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

       3. *-rgt-identity N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

       6. unpow2 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, \sqrt{\frac{1}{2}}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

       7. lower-*.f64 19.4

          \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   13. Applied rewrites 19.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   if 2e-8 < (/.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 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.9%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right)}, th\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-172}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(ky \cdot \sqrt{2}\right) \cdot \frac{\mathsf{fma}\left(kx \cdot kx, \frac{0.08333333333333333}{\sqrt{0.5}}, \sqrt{0.5}\right)}{kx}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 44.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \frac{1}{\sin kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-8)
   (* (sin th) (* ky (/ 1.0 (sin kx))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-8) {
		tmp = sin(th) * (ky * (1.0 / sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 2d-8) then
        tmp = sin(th) * (ky * (1.0d0 / sin(kx)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 2e-8) {
		tmp = Math.sin(th) * (ky * (1.0 / Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 2e-8:
		tmp = math.sin(th) * (ky * (1.0 / math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-8)
		tmp = Float64(sin(th) * Float64(ky * Float64(1.0 / sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-8)
		tmp = sin(th) * (ky * (1.0 / sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[Sin[th], $MachinePrecision] * N[(ky * N[(1.0 / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(-1 \cdot \left({ky}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right) + \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(ky \cdot \left(-1 \cdot \left({ky}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right) + \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]

      2. mul-1-neg N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{\left(\mathsf{neg}\left({ky}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right)\right)} + \frac{1}{\sin kx}\right)\right) \cdot \sin th \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \left(ky \cdot \left(\color{blue}{{ky}^{2} \cdot \left(\mathsf{neg}\left(\left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right)\right)} + \frac{1}{\sin kx}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(ky \cdot \color{blue}{\mathsf{fma}\left({ky}^{2}, \mathsf{neg}\left(\left(\frac{1}{6} \cdot \frac{1}{\sin kx} + \frac{1}{2} \cdot \frac{1}{{\sin kx}^{3}}\right)\right), \frac{1}{\sin kx}\right)}\right) \cdot \sin th \]

   5. Applied rewrites 23.0%

      \[\leadsto \color{blue}{\left(ky \cdot \mathsf{fma}\left(ky \cdot ky, \frac{-0.5}{{\sin kx}^{3}} + \frac{-0.16666666666666666}{\sin kx}, \frac{1}{\sin kx}\right)\right)} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{\sin kx}}\right) \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 25.7%

      \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{\sin kx}}\right) \cdot \sin th \]

   if 2e-8 < (/.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 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left(ky \cdot \frac{1}{\sin kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 44.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-8)
   (* (sin th) (/ ky (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-8) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 2d-8) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 2e-8) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 2e-8:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-8)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-8)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 25.7

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 25.7%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 2e-8 < (/.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 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 44.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-8)
   (/ (* ky (sin th)) (sin kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-8) {
		tmp = (ky * sin(th)) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 2d-8) then
        tmp = (ky * sin(th)) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 2e-8) {
		tmp = (ky * Math.sin(th)) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 2e-8:
		tmp = (ky * math.sin(th)) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-8)
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-8)
		tmp = (ky * sin(th)) / sin(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{\sin kx} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\sin th}}{\sin kx} \]

      4. lower-sin.f64 24.4

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sin kx}} \]

   5. Applied rewrites 24.4%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

   if 2e-8 < (/.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 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 36.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left(\left(ky \cdot \sqrt{2}\right) \cdot \frac{\sqrt{0.5}}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-8)
   (* (sin th) (* (* ky (sqrt 2.0)) (/ (sqrt 0.5) kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-8) {
		tmp = sin(th) * ((ky * sqrt(2.0)) * (sqrt(0.5) / kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 2d-8) then
        tmp = sin(th) * ((ky * sqrt(2.0d0)) * (sqrt(0.5d0) / kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 2e-8) {
		tmp = Math.sin(th) * ((ky * Math.sqrt(2.0)) * (Math.sqrt(0.5) / kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 2e-8:
		tmp = math.sin(th) * ((ky * math.sqrt(2.0)) * (math.sqrt(0.5) / kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-8)
		tmp = Float64(sin(th) * Float64(Float64(ky * sqrt(2.0)) * Float64(sqrt(0.5) / kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-8)
		tmp = sin(th) * ((ky * sqrt(2.0)) * (sqrt(0.5) / kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 36.2

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 36.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\frac{\sqrt{\frac{1}{2}}}{kx} \cdot \left(\color{blue}{ky} \cdot \sqrt{2}\right)\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 16.1%

       \[\leadsto \left(\frac{\sqrt{0.5}}{kx} \cdot \left(\color{blue}{ky} \cdot \sqrt{2}\right)\right) \cdot \sin th \]

   if 2e-8 < (/.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 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left(\left(ky \cdot \sqrt{2}\right) \cdot \frac{\sqrt{0.5}}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 36.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \frac{ky}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-8)
   (* (sin th) (* (* (sqrt 2.0) (sqrt 0.5)) (/ ky kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-8) {
		tmp = sin(th) * ((sqrt(2.0) * sqrt(0.5)) * (ky / kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 2d-8) then
        tmp = sin(th) * ((sqrt(2.0d0) * sqrt(0.5d0)) * (ky / kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 2e-8) {
		tmp = Math.sin(th) * ((Math.sqrt(2.0) * Math.sqrt(0.5)) * (ky / kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 2e-8:
		tmp = math.sin(th) * ((math.sqrt(2.0) * math.sqrt(0.5)) * (ky / kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-8)
		tmp = Float64(sin(th) * Float64(Float64(sqrt(2.0) * sqrt(0.5)) * Float64(ky / kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-8)
		tmp = sin(th) * ((sqrt(2.0) * sqrt(0.5)) * (ky / kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[Sin[th], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 36.2

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 36.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 16.1%

       \[\leadsto \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \color{blue}{\frac{ky}{kx}}\right) \cdot \sin th \]

   if 2e-8 < (/.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 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin th \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \frac{ky}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 15.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 8 \cdot 10^{-297}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      8e-297)
   (* -0.16666666666666666 (* th (* th th)))
   (fma
    th
    (* (* th th) (fma 0.008333333333333333 (* th th) -0.16666666666666666))
    th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(th) * (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 8e-297) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(th) * Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 8e-297)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 8e-297], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 8.00000000000000032e-297

   1. Initial program 96.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 21.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.3%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.7%

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   if 8.00000000000000032e-297 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 22.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 22.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.4%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right)}, th\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 8 \cdot 10^{-297}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 5e-9)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx))))
   (*
    (sin ky)
    (/
     (sin th)
     (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) <= 5e-9) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	} else {
		tmp = sin(ky) * (sin(th) / 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) <= 5e-9)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky))))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 5e-9], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.0000000000000001e-9

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.9

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 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. Applied rewrites 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 5.0000000000000001e-9 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.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. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 5e-9)
   (*
    (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) <= 5e-9) {
		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) <= 5e-9)
		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(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], 5e-9], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $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[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.0000000000000001e-9

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.9

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 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. Applied rewrites 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 5.0000000000000001e-9 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.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. unpow2 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

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      6. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      7. div-inv N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}}} \cdot \sin th \]

      10. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(kx - kx\right) - \cos \color{blue}{\left(2 \cdot kx\right)}, \frac{1}{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      11. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos kx \cdot \cos kx + \sin kx \cdot \sin kx\right)} - \cos \left(2 \cdot kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      12. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot kx\right), \frac{1}{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, \frac{1}{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      14. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(kx + kx\right)}, \frac{1}{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(kx + kx\right)}, \frac{1}{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      16. lower-+.f64 99.0

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(kx + kx\right)}, 0.5, {\sin ky}^{2}\right)}} \cdot \sin th \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \sin th \]

      18. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\sin ky \cdot \sin ky}\right)}} \cdot \sin th \]

      19. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\sin ky} \cdot \sin ky\right)}} \cdot \sin th \]

      20. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \sin ky \cdot \color{blue}{\sin ky}\right)}} \cdot \sin th \]

      21. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right)}} \cdot \sin th \]

      22. cancel-sign-sub-inv N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot ky\right)}\right)}} \cdot \sin th \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 15.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 8 \cdot 10^{-297}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      8e-297)
   (* -0.16666666666666666 (* th (* th th)))
   (fma th (* -0.16666666666666666 (* th th)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(th) * (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 8e-297) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(th) * Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 8e-297)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 8e-297], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 8.00000000000000032e-297

   1. Initial program 96.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 21.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.3%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.7%

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   if 8.00000000000000032e-297 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 22.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 22.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.5%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 8 \cdot 10^{-297}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 32: 11.0% accurate, 39.5× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* -0.16666666666666666 (* th (* th th))))

C

double code(double kx, double ky, double th) {
	return -0.16666666666666666 * (th * (th * th));
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (-0.16666666666666666d0) * (th * (th * th))
end function

Java

public static double code(double kx, double ky, double th) {
	return -0.16666666666666666 * (th * (th * th));
}

Python

def code(kx, ky, th):
	return -0.16666666666666666 * (th * (th * th))

Julia

function code(kx, ky, th)
	return Float64(-0.16666666666666666 * Float64(th * Float64(th * th)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = -0.16666666666666666 * (th * (th * th));
end

Wolfram

code[kx_, ky_, th_] := N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 21.8

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 21.8%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 13.3%

   \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

9. Taylor expanded in th around inf

   \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
10. Step-by-step derivation

11. Applied rewrites 8.4%

    \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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)))