Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 13.3s

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: 94.1% 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 95.9%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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: 82.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \sin th \cdot \frac{\sin ky}{\sqrt{t\_2 + kx \cdot kx}}\\ t_4 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_4 \leq -0.9996:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -0.3:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;t\_4 \leq 0.06:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_1 + ky \cdot ky}}\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (* (sin th) (/ (sin ky) (sqrt (+ t_2 (* kx kx))))))
        (t_4 (/ (sin ky) (sqrt (+ t_1 t_2)))))
   (if (<= t_4 -0.9996)
     t_3
     (if (<= t_4 -0.3)
       (*
        (/ (sin ky) (hypot (sin ky) (sin kx)))
        (fma th (* -0.16666666666666666 (* th th)) th))
       (if (<= t_4 0.06)
         (* (sin th) (/ (sin ky) (sqrt (+ t_1 (* ky ky)))))
         (if (<= t_4 0.995)
           (*
            (*
             (sin ky)
             (fma
              th
              (*
               (* th th)
               (fma 0.008333333333333333 (* th th) -0.16666666666666666))
              th))
            (sqrt
             (/
              1.0
              (fma
               (- 1.0 (cos (+ kx kx)))
               0.5
               (+ 0.5 (* -0.5 (cos (+ ky ky))))))))
           (if (<= t_4 2.0)
             t_3
             (*
              (sin th)
              (/
               (sin ky)
               (hypot
                (fma ky (* (* ky ky) -0.16666666666666666) ky)
                (sin kx)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(th) * (sin(ky) / sqrt((t_2 + (kx * kx))));
	double t_4 = sin(ky) / sqrt((t_1 + t_2));
	double tmp;
	if (t_4 <= -0.9996) {
		tmp = t_3;
	} else if (t_4 <= -0.3) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else if (t_4 <= 0.06) {
		tmp = sin(th) * (sin(ky) / sqrt((t_1 + (ky * ky))));
	} else if (t_4 <= 0.995) {
		tmp = (sin(ky) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th)) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))))));
	} else if (t_4 <= 2.0) {
		tmp = t_3;
	} else {
		tmp = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(t_2 + Float64(kx * kx)))))
	t_4 = Float64(sin(ky) / sqrt(Float64(t_1 + t_2)))
	tmp = 0.0
	if (t_4 <= -0.9996)
		tmp = t_3;
	elseif (t_4 <= -0.3)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	elseif (t_4 <= 0.06)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(t_1 + Float64(ky * ky)))));
	elseif (t_4 <= 0.995)
		tmp = Float64(Float64(sin(ky) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th)) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky))))))));
	elseif (t_4 <= 2.0)
		tmp = t_3;
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))));
	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[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.9996], t$95$3, If[LessEqual[t$95$4, -0.3], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.06], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], t$95$3, N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

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))))) < -0.99960000000000004 or 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 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

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

      1. unpow2 N/A

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

      2. lower-*.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -0.99960000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.299999999999999989

   1. Initial program 99.2%

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

      1. lift-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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 49.0

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

   7. Applied rewrites 49.0%

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

   if -0.299999999999999989 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.059999999999999998

   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 96.1

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

   5. Applied rewrites 96.1%

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

   if 0.059999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.6%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 74.2

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

   7. Applied rewrites 74.2%

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

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

   1. Initial program 2.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9996:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + kx \cdot kx}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.3:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.06:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + kx \cdot kx}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ t_2 := {\sin ky}^{2}\\ t_3 := \sin th \cdot \frac{\sin ky}{\sqrt{t\_2 + kx \cdot kx}}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_4 \leq -0.9996:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -0.3:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;t\_4 \leq 0.001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin kx)))))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (* (sin th) (/ (sin ky) (sqrt (+ t_2 (* kx kx))))))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_4 -0.9996)
     t_3
     (if (<= t_4 -0.3)
       (*
        (/ (sin ky) (hypot (sin ky) (sin kx)))
        (fma th (* -0.16666666666666666 (* th th)) th))
       (if (<= t_4 0.001)
         t_1
         (if (<= t_4 0.995)
           (*
            (*
             (sin ky)
             (fma
              th
              (*
               (* th th)
               (fma 0.008333333333333333 (* th th) -0.16666666666666666))
              th))
            (sqrt
             (/
              1.0
              (fma
               (- 1.0 (cos (+ kx kx)))
               0.5
               (+ 0.5 (* -0.5 (cos (+ ky ky))))))))
           (if (<= t_4 2.0) t_3 t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(th) * (sin(ky) / sqrt((t_2 + (kx * kx))));
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_4 <= -0.9996) {
		tmp = t_3;
	} else if (t_4 <= -0.3) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else if (t_4 <= 0.001) {
		tmp = t_1;
	} else if (t_4 <= 0.995) {
		tmp = (sin(ky) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th)) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))))));
	} else if (t_4 <= 2.0) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(t_2 + Float64(kx * kx)))))
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_4 <= -0.9996)
		tmp = t_3;
	elseif (t_4 <= -0.3)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	elseif (t_4 <= 0.001)
		tmp = t_1;
	elseif (t_4 <= 0.995)
		tmp = Float64(Float64(sin(ky) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th)) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky))))))));
	elseif (t_4 <= 2.0)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.9996], t$95$3, If[LessEqual[t$95$4, -0.3], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.001], t$95$1, If[LessEqual[t$95$4, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], t$95$3, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99960000000000004 or 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 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

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

      1. unpow2 N/A

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

      2. lower-*.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -0.99960000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.299999999999999989

   1. Initial program 99.2%

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

      1. lift-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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 49.0

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

   7. Applied rewrites 49.0%

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

   if -0.299999999999999989 < (/.f64 (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 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.0

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

   7. Applied rewrites 97.0%

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

   if 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))))) < 0.994999999999999996

   1. Initial program 99.6%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

      \[\leadsto \left(\sin ky \cdot \color{blue}{\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)}\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9996:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + kx \cdot kx}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.3:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin ky}^{2} + kx \cdot kx}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.9996:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_3 \leq -0.3:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;t\_3 \leq 0.001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin kx)))))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.9996)
     (* (* (sin ky) (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_3 -0.3)
       (*
        (/ (sin ky) (hypot (sin ky) (sin kx)))
        (fma th (* -0.16666666666666666 (* th th)) th))
       (if (<= t_3 0.001)
         t_1
         (if (<= t_3 0.995)
           (*
            (*
             (sin ky)
             (fma
              th
              (*
               (* th th)
               (fma 0.008333333333333333 (* th th) -0.16666666666666666))
              th))
            (sqrt
             (/
              1.0
              (fma
               (- 1.0 (cos (+ kx kx)))
               0.5
               (+ 0.5 (* -0.5 (cos (+ ky ky))))))))
           (if (<= t_3 2.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_2) 1.0))
             t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.9996) {
		tmp = (sin(ky) * sin(th)) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_3 <= -0.3) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else if (t_3 <= 0.001) {
		tmp = t_1;
	} else if (t_3 <= 0.995) {
		tmp = (sin(ky) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th)) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))))));
	} else if (t_3 <= 2.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_2), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.9996)
		tmp = Float64(Float64(sin(ky) * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_3 <= -0.3)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	elseif (t_3 <= 0.001)
		tmp = t_1;
	elseif (t_3 <= 0.995)
		tmp = Float64(Float64(sin(ky) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th)) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky))))))));
	elseif (t_3 <= 2.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_2), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.9996], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.3], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.001], t$95$1, If[LessEqual[t$95$3, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

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))))) < -0.99960000000000004

   1. Initial program 90.2%

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 54.9

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

   7. Applied rewrites 54.9%

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

   if -0.99960000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.299999999999999989

   1. Initial program 99.2%

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

      1. lift-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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 49.0

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

   7. Applied rewrites 49.0%

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

   if -0.299999999999999989 < (/.f64 (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 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.0

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

   7. Applied rewrites 97.0%

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

   if 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))))) < 0.994999999999999996

   1. Initial program 99.6%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 98.1

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9996:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.3:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ t_2 := 1 - \cos \left(kx + kx\right)\\ t_3 := \cos \left(ky + ky\right)\\ t_4 := {\sin ky}^{2}\\ t_5 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_4}}\\ \mathbf{if}\;t\_5 \leq -0.9996:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_5 \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t\_2, 0.5, \mathsf{fma}\left(t\_3, -0.5, 0.5\right)\right)}}\right)\\ \mathbf{elif}\;t\_5 \leq 0.001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t\_2, 0.5, 0.5 + -0.5 \cdot t\_3\right)}}\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_4}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (sin th)
          (/
           (sin ky)
           (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin kx)))))
        (t_2 (- 1.0 (cos (+ kx kx))))
        (t_3 (cos (+ ky ky)))
        (t_4 (pow (sin ky) 2.0))
        (t_5 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_4)))))
   (if (<= t_5 -0.9996)
     (* (* (sin ky) (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_5 -0.3)
       (*
        (fma th (* -0.16666666666666666 (* th th)) th)
        (* (sin ky) (sqrt (/ 1.0 (fma t_2 0.5 (fma t_3 -0.5 0.5))))))
       (if (<= t_5 0.001)
         t_1
         (if (<= t_5 0.995)
           (*
            (*
             (sin ky)
             (fma
              th
              (*
               (* th th)
               (fma 0.008333333333333333 (* th th) -0.16666666666666666))
              th))
            (sqrt (/ 1.0 (fma t_2 0.5 (+ 0.5 (* -0.5 t_3))))))
           (if (<= t_5 2.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_4) 1.0))
             t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	double t_2 = 1.0 - cos((kx + kx));
	double t_3 = cos((ky + ky));
	double t_4 = pow(sin(ky), 2.0);
	double t_5 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_4));
	double tmp;
	if (t_5 <= -0.9996) {
		tmp = (sin(ky) * sin(th)) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_5 <= -0.3) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) * sqrt((1.0 / fma(t_2, 0.5, fma(t_3, -0.5, 0.5)))));
	} else if (t_5 <= 0.001) {
		tmp = t_1;
	} else if (t_5 <= 0.995) {
		tmp = (sin(ky) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th)) * sqrt((1.0 / fma(t_2, 0.5, (0.5 + (-0.5 * t_3)))));
	} else if (t_5 <= 2.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_4), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))))
	t_2 = Float64(1.0 - cos(Float64(kx + kx)))
	t_3 = cos(Float64(ky + ky))
	t_4 = sin(ky) ^ 2.0
	t_5 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_4)))
	tmp = 0.0
	if (t_5 <= -0.9996)
		tmp = Float64(Float64(sin(ky) * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_5 <= -0.3)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(t_2, 0.5, fma(t_3, -0.5, 0.5))))));
	elseif (t_5 <= 0.001)
		tmp = t_1;
	elseif (t_5 <= 0.995)
		tmp = Float64(Float64(sin(ky) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th)) * sqrt(Float64(1.0 / fma(t_2, 0.5, Float64(0.5 + Float64(-0.5 * t_3))))));
	elseif (t_5 <= 2.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_4), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -0.9996], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -0.3], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(t$95$2 * 0.5 + N[(t$95$3 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.001], t$95$1, If[LessEqual[t$95$5, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(t$95$2 * 0.5 + N[(0.5 + N[(-0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$4), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]]

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))))) < -0.99960000000000004

   1. Initial program 90.2%

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 54.9

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

   7. Applied rewrites 54.9%

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

   if -0.99960000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.299999999999999989

   1. Initial program 99.2%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 48.9

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

   7. Applied rewrites 48.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. flip-+ N/A

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

      12. lift-+.f64 N/A

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

      13. lower-fma.f64 10.8

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. +-inverses N/A

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

      20. flip-+ N/A

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

      21. lift-+.f64 48.9

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

   9. Applied rewrites 48.9%

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

   if -0.299999999999999989 < (/.f64 (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 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 97.0

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

   7. Applied rewrites 97.0%

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

   if 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))))) < 0.994999999999999996

   1. Initial program 99.6%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 98.1

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9996:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \cos \left(kx + kx\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ t_4 := \cos \left(ky + ky\right)\\ \mathbf{if}\;t\_3 \leq -0.9996:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t\_1, 0.5, \mathsf{fma}\left(t\_4, -0.5, 0.5\right)\right)}}\right)\\ \mathbf{elif}\;t\_3 \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}{\sqrt{t\_2 + ky \cdot ky}}\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t\_1, 0.5, 0.5 + -0.5 \cdot t\_4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (- 1.0 (cos (+ kx kx))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0)))))
        (t_4 (cos (+ ky ky))))
   (if (<= t_3 -0.9996)
     (* (* (sin ky) (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_3 -0.2)
       (*
        (fma th (* -0.16666666666666666 (* th th)) th)
        (* (sin ky) (sqrt (/ 1.0 (fma t_1 0.5 (fma t_4 -0.5 0.5))))))
       (if (<= t_3 0.001)
         (*
          (sin th)
          (/
           (fma ky (* (* ky ky) -0.16666666666666666) ky)
           (sqrt (+ t_2 (* ky ky)))))
         (if (<= t_3 0.995)
           (*
            (*
             (sin ky)
             (fma
              th
              (*
               (* th th)
               (fma 0.008333333333333333 (* th th) -0.16666666666666666))
              th))
            (sqrt (/ 1.0 (fma t_1 0.5 (+ 0.5 (* -0.5 t_4))))))
           (sin th)))))))

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 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double t_4 = cos((ky + ky));
	double tmp;
	if (t_3 <= -0.9996) {
		tmp = (sin(ky) * sin(th)) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_3 <= -0.2) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) * sqrt((1.0 / fma(t_1, 0.5, fma(t_4, -0.5, 0.5)))));
	} else if (t_3 <= 0.001) {
		tmp = sin(th) * (fma(ky, ((ky * ky) * -0.16666666666666666), ky) / sqrt((t_2 + (ky * ky))));
	} else if (t_3 <= 0.995) {
		tmp = (sin(ky) * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th)) * sqrt((1.0 / fma(t_1, 0.5, (0.5 + (-0.5 * t_4)))));
	} else {
		tmp = sin(th);
	}
	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 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	t_4 = cos(Float64(ky + ky))
	tmp = 0.0
	if (t_3 <= -0.9996)
		tmp = Float64(Float64(sin(ky) * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_3 <= -0.2)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(t_1, 0.5, fma(t_4, -0.5, 0.5))))));
	elseif (t_3 <= 0.001)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky) / sqrt(Float64(t_2 + Float64(ky * ky)))));
	elseif (t_3 <= 0.995)
		tmp = Float64(Float64(sin(ky) * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th)) * sqrt(Float64(1.0 / fma(t_1, 0.5, Float64(0.5 + Float64(-0.5 * t_4))))));
	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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -0.9996], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(t$95$1 * 0.5 + N[(t$95$4 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.001], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(t$95$1 * 0.5 + N[(0.5 + N[(-0.5 * t$95$4), $MachinePrecision]), $MachinePrecision]), $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))))) < -0.99960000000000004

   1. Initial program 90.2%

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 54.9

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

   7. Applied rewrites 54.9%

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

   if -0.99960000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.2%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.6

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

   7. Applied rewrites 45.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. flip-+ N/A

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

      12. lift-+.f64 N/A

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

      13. lower-fma.f64 10.1

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. +-inverses N/A

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

      20. flip-+ N/A

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

      21. lift-+.f64 45.6

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

   9. Applied rewrites 45.6%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1e-3

   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 98.7

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

   5. Applied rewrites 98.7%

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

   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 98.6

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

   8. Applied rewrites 98.6%

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

   if 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))))) < 0.994999999999999996

   1. Initial program 99.6%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9996:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.9996:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{t\_1}}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\right)\\ \mathbf{elif}\;t\_3 \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}{\sqrt{t\_2 + ky \cdot ky}}\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fma -0.5 (cos (* ky -2.0)) 0.5))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.9996)
     (* (* (sin ky) (sin th)) (sqrt (/ 1.0 t_1)))
     (if (<= t_3 -0.2)
       (*
        (fma th (* -0.16666666666666666 (* th th)) th)
        (*
         (sin ky)
         (sqrt
          (/
           1.0
           (fma (- 1.0 (cos (+ kx kx))) 0.5 (fma (cos (+ ky ky)) -0.5 0.5))))))
       (if (<= t_3 0.001)
         (*
          (sin th)
          (/
           (fma ky (* (* ky ky) -0.16666666666666666) ky)
           (sqrt (+ t_2 (* ky ky)))))
         (if (<= t_3 0.995)
           (*
            (* (sin ky) th)
            (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_1))))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma(-0.5, cos((ky * -2.0)), 0.5);
	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 <= -0.9996) {
		tmp = (sin(ky) * sin(th)) * sqrt((1.0 / t_1));
	} else if (t_3 <= -0.2) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, fma(cos((ky + ky)), -0.5, 0.5)))));
	} else if (t_3 <= 0.001) {
		tmp = sin(th) * (fma(ky, ((ky * ky) * -0.16666666666666666), ky) / sqrt((t_2 + (ky * ky))));
	} else if (t_3 <= 0.995) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_1)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	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 <= -0.9996)
		tmp = Float64(Float64(sin(ky) * sin(th)) * sqrt(Float64(1.0 / t_1)));
	elseif (t_3 <= -0.2)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, fma(cos(Float64(ky + ky)), -0.5, 0.5))))));
	elseif (t_3 <= 0.001)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky) / sqrt(Float64(t_2 + Float64(ky * ky)))));
	elseif (t_3 <= 0.995)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_1))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $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, -0.9996], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.001], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $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))))) < -0.99960000000000004

   1. Initial program 90.2%

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 54.9

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

   7. Applied rewrites 54.9%

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

   if -0.99960000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.2%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.6

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

   7. Applied rewrites 45.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. flip-+ N/A

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

      12. lift-+.f64 N/A

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

      13. lower-fma.f64 10.1

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

      14. lift-+.f64 N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. +-inverses N/A

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

      20. flip-+ N/A

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

      21. lift-+.f64 45.6

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

   9. Applied rewrites 45.6%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1e-3

   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 98.7

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

   5. Applied rewrites 98.7%

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

   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 98.6

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

   8. Applied rewrites 98.6%

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

   if 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))))) < 0.994999999999999996

   1. Initial program 99.6%

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

   4. Applied rewrites 99.1%

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

   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 75.7%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9996:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ t_2 := \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_1\right)}}\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_4 \leq -0.9996:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{t\_1}}\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}{\sqrt{t\_3 + ky \cdot ky}}\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fma -0.5 (cos (* ky -2.0)) 0.5))
        (t_2
         (*
          (* (sin ky) th)
          (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_1)))))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin ky) 2.0))))))
   (if (<= t_4 -0.9996)
     (* (* (sin ky) (sin th)) (sqrt (/ 1.0 t_1)))
     (if (<= t_4 -0.2)
       t_2
       (if (<= t_4 0.001)
         (*
          (sin th)
          (/
           (fma ky (* (* ky ky) -0.16666666666666666) ky)
           (sqrt (+ t_3 (* ky ky)))))
         (if (<= t_4 0.995) t_2 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double t_2 = (sin(ky) * th) * sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_1)));
	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 <= -0.9996) {
		tmp = (sin(ky) * sin(th)) * sqrt((1.0 / t_1));
	} else if (t_4 <= -0.2) {
		tmp = t_2;
	} else if (t_4 <= 0.001) {
		tmp = sin(th) * (fma(ky, ((ky * ky) * -0.16666666666666666), ky) / sqrt((t_3 + (ky * ky))));
	} else if (t_4 <= 0.995) {
		tmp = t_2;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	t_2 = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_1))))
	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 <= -0.9996)
		tmp = Float64(Float64(sin(ky) * sin(th)) * sqrt(Float64(1.0 / t_1)));
	elseif (t_4 <= -0.2)
		tmp = t_2;
	elseif (t_4 <= 0.001)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky) / sqrt(Float64(t_3 + Float64(ky * ky)))));
	elseif (t_4 <= 0.995)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $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, -0.9996], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.2], t$95$2, If[LessEqual[t$95$4, 0.001], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], t$95$2, 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))))) < -0.99960000000000004

   1. Initial program 90.2%

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 54.9

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

   7. Applied rewrites 54.9%

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

   if -0.99960000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 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))))) < 0.994999999999999996

   1. Initial program 99.4%

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

   4. Applied rewrites 99.1%

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

   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 58.8%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1e-3

   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 98.7

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

   5. Applied rewrites 98.7%

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

   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 98.6

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

   8. Applied rewrites 98.6%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9996:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.9:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{t\_1}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.9)
     (* (* (sin ky) th) (sqrt (/ 1.0 t_1)))
     (if (<= t_2 2e-286)
       (* (* (sin ky) (sin th)) (sqrt (/ 2.0 (- 1.0 (cos (* kx -2.0))))))
       (if (<= t_2 4e-7) (* (sin th) (/ (sin ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.9) {
		tmp = (sin(ky) * th) * sqrt((1.0 / t_1));
	} else if (t_2 <= 2e-286) {
		tmp = (sin(ky) * sin(th)) * sqrt((2.0 / (1.0 - cos((kx * -2.0)))));
	} else if (t_2 <= 4e-7) {
		tmp = sin(th) * (sin(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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(ky) ** 2.0d0
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + t_1))
    if (t_2 <= (-0.9d0)) then
        tmp = (sin(ky) * th) * sqrt((1.0d0 / t_1))
    else if (t_2 <= 2d-286) then
        tmp = (sin(ky) * sin(th)) * sqrt((2.0d0 / (1.0d0 - cos((kx * (-2.0d0))))))
    else if (t_2 <= 4d-7) then
        tmp = sin(th) * (sin(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 t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.9) {
		tmp = (Math.sin(ky) * th) * Math.sqrt((1.0 / t_1));
	} else if (t_2 <= 2e-286) {
		tmp = (Math.sin(ky) * Math.sin(th)) * Math.sqrt((2.0 / (1.0 - Math.cos((kx * -2.0)))));
	} else if (t_2 <= 4e-7) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	tmp = 0
	if t_2 <= -0.9:
		tmp = (math.sin(ky) * th) * math.sqrt((1.0 / t_1))
	elif t_2 <= 2e-286:
		tmp = (math.sin(ky) * math.sin(th)) * math.sqrt((2.0 / (1.0 - math.cos((kx * -2.0)))))
	elif t_2 <= 4e-7:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.9)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / t_1)));
	elseif (t_2 <= 2e-286)
		tmp = Float64(Float64(sin(ky) * sin(th)) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(kx * -2.0))))));
	elseif (t_2 <= 4e-7)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	tmp = 0.0;
	if (t_2 <= -0.9)
		tmp = (sin(ky) * th) * sqrt((1.0 / t_1));
	elseif (t_2 <= 2e-286)
		tmp = (sin(ky) * sin(th)) * sqrt((2.0 / (1.0 - cos((kx * -2.0)))));
	elseif (t_2 <= 4e-7)
		tmp = sin(th) * (sin(ky) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.9], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-286], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-7], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.900000000000000022

   1. Initial program 91.8%

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

   3. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 40.7%

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

   if -0.900000000000000022 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.0000000000000001e-286

   1. Initial program 99.4%

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

   4. Applied rewrites 81.8%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower--.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.1

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

   7. Applied rewrites 59.1%

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

   if 2.0000000000000001e-286 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-7

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 64.3

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

   5. Applied rewrites 64.3%

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

   if 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_2 \leq 0.06:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}{\sqrt{t\_1 + ky \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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))))))
   (if (<= t_2 -0.2)
     (* (* (sin ky) (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_2 0.06)
       (*
        (sin th)
        (/
         (fma ky (* (* ky ky) -0.16666666666666666) ky)
         (sqrt (+ t_1 (* ky ky)))))
       (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 tmp;
	if (t_2 <= -0.2) {
		tmp = (sin(ky) * sin(th)) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_2 <= 0.06) {
		tmp = sin(th) * (fma(ky, ((ky * ky) * -0.16666666666666666), ky) / sqrt((t_1 + (ky * ky))));
	} 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))))
	tmp = 0.0
	if (t_2 <= -0.2)
		tmp = Float64(Float64(sin(ky) * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_2 <= 0.06)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky) / sqrt(Float64(t_1 + Float64(ky * ky)))));
	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]}, If[LessEqual[t$95$2, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.06], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $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))))) < -0.20000000000000001

   1. Initial program 94.0%

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 41.3

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

   7. Applied rewrites 41.3%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.059999999999999998

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.8

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

   5. Applied rewrites 97.8%

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

   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 0.059999999999999998 < (/.f64 (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 92.7%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 72.1

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

   5. Applied rewrites 72.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.06:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (* (* (sin ky) (sin th)) (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 4e-7) (* (sin ky) (/ (sin th) (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (sin(ky) * sin(th)) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 4e-7) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(Float64(sin(ky) * sin(th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 4e-7)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-7], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.0%

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 41.3

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

   7. Applied rewrites 41.3%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 65.3

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

   5. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 65.3

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

   7. Applied rewrites 65.3%

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

   if 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.9:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{t\_1}}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.9)
     (* (* (sin ky) th) (sqrt (/ 1.0 t_1)))
     (if (<= t_2 4e-7) (* (sin ky) (/ (sin th) (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.9) {
		tmp = (sin(ky) * th) * sqrt((1.0 / t_1));
	} else if (t_2 <= 4e-7) {
		tmp = sin(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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(ky) ** 2.0d0
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + t_1))
    if (t_2 <= (-0.9d0)) then
        tmp = (sin(ky) * th) * sqrt((1.0d0 / t_1))
    else if (t_2 <= 4d-7) then
        tmp = sin(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 t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.9) {
		tmp = (Math.sin(ky) * th) * Math.sqrt((1.0 / t_1));
	} else if (t_2 <= 4e-7) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	tmp = 0
	if t_2 <= -0.9:
		tmp = (math.sin(ky) * th) * math.sqrt((1.0 / t_1))
	elif t_2 <= 4e-7:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.9)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / t_1)));
	elseif (t_2 <= 4e-7)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	tmp = 0.0;
	if (t_2 <= -0.9)
		tmp = (sin(ky) * th) * sqrt((1.0 / t_1));
	elseif (t_2 <= 4e-7)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.9], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-7], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.900000000000000022

   1. Initial program 91.8%

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

   3. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 40.7%

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

   if -0.900000000000000022 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 56.7

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

   5. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 56.7

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

   7. Applied rewrites 56.7%

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

   if 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.9:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(ky \cdot -2\right), -0.5, 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.9)
     (*
      (fma th (* -0.16666666666666666 (* th th)) th)
      (* (sin ky) (sqrt (/ 1.0 (fma (cos (* ky -2.0)) -0.5 0.5)))))
     (if (<= t_1 4e-7) (* (sin ky) (/ (sin th) (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 <= -0.9) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) * sqrt((1.0 / fma(cos((ky * -2.0)), -0.5, 0.5))));
	} else if (t_1 <= 4e-7) {
		tmp = sin(ky) * (sin(th) / 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 <= -0.9)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(cos(Float64(ky * -2.0)), -0.5, 0.5)))));
	elseif (t_1 <= 4e-7)
		tmp = Float64(sin(ky) * Float64(sin(th) / 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, -0.9], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-7], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.900000000000000022

   1. Initial program 91.8%

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.9

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

   7. Applied rewrites 36.9%

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

   8. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 28.1

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

   10. Applied rewrites 28.1%

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

   if -0.900000000000000022 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 56.7

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

   5. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 56.7

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

   7. Applied rewrites 56.7%

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

   if 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(ky \cdot -2\right), -0.5, 0.5\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(ky \cdot -2\right), -0.5, 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.3)
     (*
      (fma th (* -0.16666666666666666 (* th th)) th)
      (* (sin ky) (sqrt (/ 1.0 (fma (cos (* ky -2.0)) -0.5 0.5)))))
     (if (<= t_1 4e-7) (* (sin th) (/ ky (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.3) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) * sqrt((1.0 / fma(cos((ky * -2.0)), -0.5, 0.5))));
	} else if (t_1 <= 4e-7) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.3)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(cos(Float64(ky * -2.0)), -0.5, 0.5)))));
	elseif (t_1 <= 4e-7)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.3], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-7], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.299999999999999989

   1. Initial program 93.8%

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

   4. Applied rewrites 72.9%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.7

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

   7. Applied rewrites 37.7%

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

   8. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 23.1

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

   10. Applied rewrites 23.1%

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

   if -0.299999999999999989 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 63.9

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

   5. Applied rewrites 63.9%

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

   if 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(ky \cdot -2\right), -0.5, 0.5\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (* (* (sin ky) th) (sqrt (/ 1.0 (- 1.0 (cos (* ky -2.0))))))
     (if (<= t_1 4e-7) (* (sin th) (/ ky (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (sin(ky) * th) * sqrt((1.0 / (1.0 - cos((ky * -2.0)))));
	} else if (t_1 <= 4e-7) {
		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) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.2d0)) then
        tmp = (sin(ky) * th) * sqrt((1.0d0 / (1.0d0 - cos((ky * (-2.0d0))))))
    else if (t_1 <= 4d-7) 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 t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (Math.sin(ky) * th) * Math.sqrt((1.0 / (1.0 - Math.cos((ky * -2.0)))));
	} else if (t_1 <= 4e-7) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.2:
		tmp = (math.sin(ky) * th) * math.sqrt((1.0 / (1.0 - math.cos((ky * -2.0)))))
	elif t_1 <= 4e-7:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(ky * -2.0))))));
	elseif (t_1 <= 4e-7)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = (sin(ky) * th) * sqrt((1.0 / (1.0 - cos((ky * -2.0)))));
	elseif (t_1 <= 4e-7)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-7], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.0%

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

   3. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 45.2

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

   5. Applied rewrites 45.2%

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

   7. Applied rewrites 9.6%

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

   8. Taylor expanded in ky around inf

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

   10. Applied rewrites 9.5%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999998e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{1 - \cos \left(ky \cdot -2\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-7)
   (* (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)))) <= 4e-7) {
		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)))) <= 4d-7) 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)))) <= 4e-7) {
		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)))) <= 4e-7:
		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)))) <= 4e-7)
		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)))) <= 4e-7)
		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], 4e-7], 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))))) < 3.9999999999999998e-7

   1. Initial program 97.3%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 40.9

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

   5. Applied rewrites 40.9%

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

   if 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-7)
   (/ (* 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)))) <= 4e-7) {
		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)))) <= 4d-7) 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)))) <= 4e-7) {
		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)))) <= 4e-7:
		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)))) <= 4e-7)
		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)))) <= 4e-7)
		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], 4e-7], 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))))) < 3.9999999999999998e-7

   1. Initial program 97.3%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   if 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 37.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{2 \cdot \left(kx \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-7)
   (* (sin th) (* (sqrt (/ 1.0 (* 2.0 (* kx kx)))) (* ky (sqrt 2.0))))
   (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-7) {
		tmp = sin(th) * (sqrt((1.0 / (2.0 * (kx * kx)))) * (ky * sqrt(2.0)));
	} 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-7) then
        tmp = sin(th) * (sqrt((1.0d0 / (2.0d0 * (kx * kx)))) * (ky * sqrt(2.0d0)))
    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-7) {
		tmp = Math.sin(th) * (Math.sqrt((1.0 / (2.0 * (kx * kx)))) * (ky * Math.sqrt(2.0)));
	} 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-7:
		tmp = math.sin(th) * (math.sqrt((1.0 / (2.0 * (kx * kx)))) * (ky * math.sqrt(2.0)))
	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-7)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(2.0 * Float64(kx * kx)))) * Float64(ky * sqrt(2.0))));
	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-7)
		tmp = sin(th) * (sqrt((1.0 / (2.0 * (kx * kx)))) * (ky * sqrt(2.0)));
	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-7], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(2.0 * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $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))))) < 1.9999999999999999e-7

   1. Initial program 97.3%

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

   4. Applied rewrites 74.8%

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

   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 45.3

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

   7. Applied rewrites 45.3%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 29.0%

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

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

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{2 \cdot \left(kx \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \left(\left(ky \cdot \sqrt{2}\right) \cdot \frac{\sqrt{0.5}}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-7)
   (* (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-7) {
		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-7) 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-7) {
		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-7:
		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-7)
		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-7)
		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-7], 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))))) < 1.9999999999999999e-7

   1. Initial program 97.3%

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

   4. Applied rewrites 74.8%

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

   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 45.3

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

   7. Applied rewrites 45.3%

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

   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 26.1%

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

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

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \left(\left(ky \cdot \sqrt{2}\right) \cdot \frac{\sqrt{0.5}}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \frac{ky}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-7)
   (* (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-7) {
		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-7) 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-7) {
		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-7:
		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-7)
		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-7)
		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-7], 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))))) < 1.9999999999999999e-7

   1. Initial program 97.3%

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

   4. Applied rewrites 74.8%

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

   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 45.3

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

   7. Applied rewrites 45.3%

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

   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 26.1%

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

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

   1. Initial program 92.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 71.4

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

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \frac{ky}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 14.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-308}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      1e-308)
   (* -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))))) <= 1e-308) {
		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))))) <= 1e-308)
		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], 1e-308], 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)) < 9.9999999999999991e-309

   1. Initial program 96.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 24.3

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

   5. Applied rewrites 24.3%

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

   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 16.4%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 19.1%

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   if 9.9999999999999991e-309 < (*.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 95.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 28.1

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

   5. Applied rewrites 28.1%

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

   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 14.1%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right)}, th\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-308}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 14.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-308}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      1e-308)
   (* -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))))) <= 1e-308) {
		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))))) <= 1e-308)
		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], 1e-308], 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)) < 9.9999999999999991e-309

   1. Initial program 96.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 24.3

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

   5. Applied rewrites 24.3%

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

   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 16.4%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 19.1%

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   if 9.9999999999999991e-309 < (*.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 95.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 28.1

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

   5. Applied rewrites 28.1%

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

   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 14.0%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-308}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 11.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-308}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      1e-308)
   (* -0.16666666666666666 (* th (* th th)))
   (* th (sqrt 0.5))))

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))))) <= 1e-308) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = th * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(th) * (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))))) <= 1d-308) then
        tmp = (-0.16666666666666666d0) * (th * (th * th))
    else
        tmp = th * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(th) * (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))))) <= 1e-308) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = th * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(th) * (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))))) <= 1e-308:
		tmp = -0.16666666666666666 * (th * (th * th))
	else:
		tmp = th * math.sqrt(0.5)
	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))))) <= 1e-308)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = Float64(th * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(th) * (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 1e-308)
		tmp = -0.16666666666666666 * (th * (th * th));
	else
		tmp = th * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-308], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 9.9999999999999991e-309

   1. Initial program 96.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 24.3

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

   5. Applied rewrites 24.3%

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

   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 16.4%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 19.1%

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   if 9.9999999999999991e-309 < (*.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 95.0%

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

   3. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 37.9

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

   5. Applied rewrites 37.9%

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

   7. Applied rewrites 6.7%

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

   8. Taylor expanded in ky around 0

      \[\leadsto th \cdot \color{blue}{\sqrt{\frac{1}{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 7.0%

       \[\leadsto th \cdot \color{blue}{\sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-308}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th \cdot \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 35.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-7)
   (* ky (/ 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)))) <= 4e-7) {
		tmp = ky * (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)))) <= 4d-7) then
        tmp = ky * (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)))) <= 4e-7) {
		tmp = ky * (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)))) <= 4e-7:
		tmp = ky * (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)))) <= 4e-7)
		tmp = Float64(ky * Float64(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)))) <= 4e-7)
		tmp = ky * (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], 4e-7], N[(ky * N[(th / 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))))) < 3.9999999999999998e-7

   1. Initial program 97.3%

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

   3. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot th\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot th\right)} \]

      13. lower-sin.f64 47.2

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\color{blue}{\sin ky} \cdot th\right) \]

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\sin ky \cdot th\right)} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
   7. Step-by-step derivation

   8. Applied rewrites 26.8%

      \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]

   if 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 71.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 26.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\frac{1}{kx}}\\ t_2 := th \cdot \sqrt{0.5}\\ \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-24}:\\ \;\;\;\;ky \cdot \mathsf{fma}\left(-0.16666666666666666, t\_1 \cdot \left(\left(ky \cdot ky\right) \cdot t\_2\right), t\_1 \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sqrt (/ 1.0 kx))) (t_2 (* th (sqrt 0.5))))
   (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-24)
     (* ky (fma -0.16666666666666666 (* t_1 (* (* ky ky) t_2)) (* t_1 t_2)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double t_1 = sqrt((1.0 / kx));
	double t_2 = th * sqrt(0.5);
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 1e-24) {
		tmp = ky * fma(-0.16666666666666666, (t_1 * ((ky * ky) * t_2)), (t_1 * t_2));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sqrt(Float64(1.0 / kx))
	t_2 = Float64(th * sqrt(0.5))
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-24)
		tmp = Float64(ky * fma(-0.16666666666666666, Float64(t_1 * Float64(Float64(ky * ky) * t_2)), Float64(t_1 * t_2)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[(1.0 / kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(th * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]}, 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], 1e-24], N[(ky * N[(-0.16666666666666666 * N[(t$95$1 * N[(N[(ky * ky), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * t$95$2), $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))))) < 9.99999999999999924e-25

   1. Initial program 97.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(th \cdot \sin ky\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(th \cdot \sin ky\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{{\color{blue}{\sin ky}}^{2} + {\sin kx}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot th\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot th\right)} \]

      13. lower-sin.f64 47.4

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\color{blue}{\sin ky} \cdot th\right) \]

   5. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\sin ky \cdot th\right)} \]

   7. Applied rewrites 5.1%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right) + \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}} \cdot th\right) \cdot \color{blue}{\sin ky} \]
   8. Step-by-step derivation

   9. Applied rewrites 13.3%

      \[\leadsto \left(\sqrt{\frac{1}{kx + kx}} \cdot th\right) \cdot \sin ky \]

   10. Taylor expanded in ky around 0

       \[\leadsto ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sqrt{\frac{1}{kx}} \cdot \left({ky}^{2} \cdot \left(th \cdot \sqrt{\frac{1}{2}}\right)\right)\right) + \sqrt{\frac{1}{kx}} \cdot \left(th \cdot \sqrt{\frac{1}{2}}\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 12.7%

       \[\leadsto ky \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \sqrt{\frac{1}{kx}} \cdot \left(\left(ky \cdot ky\right) \cdot \left(th \cdot \sqrt{0.5}\right)\right), \sqrt{\frac{1}{kx}} \cdot \left(th \cdot \sqrt{0.5}\right)\right)} \]

   if 9.99999999999999924e-25 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 69.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 26.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right) \cdot \left(th \cdot \sqrt{\frac{1}{kx + kx}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-24)
   (*
    (fma ky (* (* ky ky) -0.16666666666666666) ky)
    (* th (sqrt (/ 1.0 (+ kx 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)))) <= 1e-24) {
		tmp = fma(ky, ((ky * ky) * -0.16666666666666666), ky) * (th * sqrt((1.0 / (kx + kx))));
	} else {
		tmp = 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)))) <= 1e-24)
		tmp = Float64(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky) * Float64(th * sqrt(Float64(1.0 / Float64(kx + kx)))));
	else
		tmp = sin(th);
	end
	return 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], 1e-24], N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] * N[(th * N[Sqrt[N[(1.0 / N[(kx + kx), $MachinePrecision]), $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))))) < 9.99999999999999924e-25

   1. Initial program 97.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(th \cdot \sin ky\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(th \cdot \sin ky\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{{\color{blue}{\sin ky}}^{2} + {\sin kx}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot th\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot th\right)} \]

      13. lower-sin.f64 47.4

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\color{blue}{\sin ky} \cdot th\right) \]

   5. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\sin ky \cdot th\right)} \]

   7. Applied rewrites 5.1%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right) + \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}} \cdot th\right) \cdot \color{blue}{\sin ky} \]
   8. Step-by-step derivation

   9. Applied rewrites 13.3%

      \[\leadsto \left(\sqrt{\frac{1}{kx + kx}} \cdot th\right) \cdot \sin ky \]

   10. Taylor expanded in ky around 0

       \[\leadsto \left(\sqrt{\frac{1}{kx + kx}} \cdot th\right) \cdot \left(ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 12.7%

       \[\leadsto \left(\sqrt{\frac{1}{kx + kx}} \cdot th\right) \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right) \cdot -0.16666666666666666}, ky\right) \]

   if 9.99999999999999924e-25 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 69.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right) \cdot \left(th \cdot \sqrt{\frac{1}{kx + kx}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 16.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right) \cdot \left(th \cdot \sqrt{\frac{1}{kx + kx}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-7)
   (*
    (fma ky (* (* ky ky) -0.16666666666666666) ky)
    (* th (sqrt (/ 1.0 (+ kx kx)))))
   (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(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-7) {
		tmp = fma(ky, ((ky * ky) * -0.16666666666666666), ky) * (th * sqrt((1.0 / (kx + kx))));
	} 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(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-7)
		tmp = Float64(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky) * Float64(th * sqrt(Float64(1.0 / Float64(kx + kx)))));
	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[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-7], N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] * N[(th * N[Sqrt[N[(1.0 / N[(kx + kx), $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]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 97.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(th \cdot \sin ky\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(th \cdot \sin ky\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{{\color{blue}{\sin ky}}^{2} + {\sin kx}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot th\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot th\right)} \]

      13. lower-sin.f64 47.2

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\color{blue}{\sin ky} \cdot th\right) \]

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\sin ky \cdot th\right)} \]

   7. Applied rewrites 5.0%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right) + \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}} \cdot th\right) \cdot \color{blue}{\sin ky} \]
   8. Step-by-step derivation

   9. Applied rewrites 13.2%

      \[\leadsto \left(\sqrt{\frac{1}{kx + kx}} \cdot th\right) \cdot \sin ky \]

   10. Taylor expanded in ky around 0

       \[\leadsto \left(\sqrt{\frac{1}{kx + kx}} \cdot th\right) \cdot \left(ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 12.6%

       \[\leadsto \left(\sqrt{\frac{1}{kx + kx}} \cdot th\right) \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right) \cdot -0.16666666666666666}, ky\right) \]

   if 1.9999999999999999e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 71.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\sin th} \]

   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 39.8%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right)}, th\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right) \cdot \left(th \cdot \sqrt{\frac{1}{kx + kx}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 16.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{1}{kx}} \cdot \left(ky \cdot \left(th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-7)
   (* (sqrt (/ 1.0 kx)) (* ky (* th (sqrt 0.5))))
   (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(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-7) {
		tmp = sqrt((1.0 / kx)) * (ky * (th * sqrt(0.5)));
	} 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(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-7)
		tmp = Float64(sqrt(Float64(1.0 / kx)) * Float64(ky * Float64(th * sqrt(0.5))));
	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[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-7], N[(N[Sqrt[N[(1.0 / kx), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(th * N[Sqrt[0.5], $MachinePrecision]), $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 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 97.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(th \cdot \sin ky\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(th \cdot \sin ky\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{{\color{blue}{\sin ky}}^{2} + {\sin kx}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \left(th \cdot \sin ky\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \left(th \cdot \sin ky\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot th\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \color{blue}{\left(\sin ky \cdot th\right)} \]

      13. lower-sin.f64 47.2

          \[\leadsto \sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\color{blue}{\sin ky} \cdot th\right) \]

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\sin ky \cdot th\right)} \]

   7. Applied rewrites 5.0%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right) + \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}} \cdot th\right) \cdot \color{blue}{\sin ky} \]
   8. Step-by-step derivation

   9. Applied rewrites 13.2%

      \[\leadsto \left(\sqrt{\frac{1}{kx + kx}} \cdot th\right) \cdot \sin ky \]

   10. Taylor expanded in ky around 0

       \[\leadsto \sqrt{\frac{1}{kx}} \cdot \color{blue}{\left(ky \cdot \left(th \cdot \sqrt{\frac{1}{2}}\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 12.7%

       \[\leadsto \sqrt{\frac{1}{kx}} \cdot \color{blue}{\left(ky \cdot \left(th \cdot \sqrt{0.5}\right)\right)} \]

   if 1.9999999999999999e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 71.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\sin th} \]

   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 39.8%

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right)}, th\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0034:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0034)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin kx))))
   (/
    (* (sin ky) (sin th))
    (sqrt
     (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0034) {
		tmp = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(kx)));
	} else {
		tmp = (sin(ky) * sin(th)) / sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0034)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(kx))));
	else
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0034], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.00339999999999999981

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}, \sin kx\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}, \sin kx\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}, \sin kx\right)} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

      8. lower-*.f64 72.2

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 72.2%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}, \sin kx\right)} \cdot \sin th \]

   if 0.00339999999999999981 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-*.f64 99.7

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      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 99.1%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0034:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0034:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0034)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (fma ky (* (* ky ky) -0.16666666666666666) ky) (sin 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 (ky <= 0.0034) {
		tmp = sin(th) * (sin(ky) / hypot(fma(ky, ((ky * ky) * -0.16666666666666666), ky), sin(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 (ky <= 0.0034)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(fma(ky, Float64(Float64(ky * ky) * -0.16666666666666666), ky), sin(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[ky, 0.0034], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[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 ky < 0.00339999999999999981

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}, \sin kx\right)} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}, \sin kx\right)} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}, \sin kx\right)} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right), \sin kx\right)} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

      8. lower-*.f64 72.2

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 72.2%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}, \sin kx\right)} \cdot \sin th \]

   if 0.00339999999999999981 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      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 99.6

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(kx + kx\right)}, 0.5, {\sin ky}^{2}\right)}} \cdot \sin th \]

      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 99.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0034:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right), \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 44.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.6 \cdot 10^{-155}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;ky \leq 0.00058:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.6e-155)
   (* (sin th) (/ ky (sin kx)))
   (if (<= ky 0.00058)
     (*
      (sin th)
      (* (sin ky) (sqrt (/ 1.0 (fma (- 1.0 (cos (+ kx kx))) 0.5 (* ky ky))))))
     (*
      (sin th)
      (* (sin ky) (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.6e-155) {
		tmp = sin(th) * (ky / sin(kx));
	} else if (ky <= 0.00058) {
		tmp = sin(th) * (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (ky * ky)))));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.6e-155)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	elseif (ky <= 0.00058)
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(ky * ky))))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5)))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.6e-155], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 0.00058], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 4.60000000000000011e-155

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 37.0

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 37.0%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 4.60000000000000011e-155 < ky < 5.8e-4

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. lower-*.f64 86.3

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Applied rewrites 86.3%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   if 5.8e-4 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right) + \frac{1}{2}}} \cdot \sin ky\right) \cdot \sin th \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{-1}{2} \cdot \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)} + \frac{1}{2}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right), \frac{1}{2}\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      5. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, \frac{1}{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, \frac{1}{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-*.f64 65.9

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(ky \cdot -2\right)}, 0.5\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Applied rewrites 65.9%

      \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 4.6 \cdot 10^{-155}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;ky \leq 0.00058:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 32: 10.4% 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 95.9%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 25.9

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 25.9%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 15.4%

   \[\leadsto \mathsf{fma}\left(th, \color{blue}{-0.16666666666666666 \cdot \left(th \cdot th\right)}, th\right) \]

9. Taylor expanded in th around inf

   \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
10. Step-by-step derivation

11. Applied rewrites 13.1%

    \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(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)))