Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 11.2s

Alternatives: 22

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 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. Final simplification 99.7%

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

Alternative 2: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \cos \left(ky + ky\right)\\ t_3 := \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{if}\;t\_1 \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{\frac{1}{\left(2 - t\_2\right) - \cos \left(2 \cdot kx\right)}}\right) \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.9972580999534978:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, -0.0001984126984126984, 0.008333333333333333\right), th \cdot th, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - t\_2, 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (cos (+ ky ky)))
        (t_3
         (*
          (/
           (sin ky)
           (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
          (sin th))))
   (if (<= t_1 -0.945)
     (* (* (/ 1.0 (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5))) (sin ky)) (sin th))
     (if (<= t_1 -0.1)
       (*
        (* (sqrt 2.0) (sqrt (/ 1.0 (- (- 2.0 t_2) (cos (* 2.0 kx))))))
        (* th (sin ky)))
       (if (<= t_1 1e-7)
         t_3
         (if (<= t_1 0.9972580999534978)
           (*
            (*
             (fma
              (fma
               (fma (* th th) -0.0001984126984126984 0.008333333333333333)
               (* th th)
               -0.16666666666666666)
              (* th th)
              1.0)
             th)
            (/
             (sin ky)
             (sqrt (fma (- 1.0 t_2) 0.5 (* (- 1.0 (cos (+ kx kx))) 0.5)))))
           (if (<= t_1 2.0) (sin th) t_3)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = cos((ky + ky));
	double t_3 = (sin(ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	double tmp;
	if (t_1 <= -0.945) {
		tmp = ((1.0 / sqrt(((1.0 - cos((2.0 * ky))) * 0.5))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = (sqrt(2.0) * sqrt((1.0 / ((2.0 - t_2) - cos((2.0 * kx)))))) * (th * sin(ky));
	} else if (t_1 <= 1e-7) {
		tmp = t_3;
	} else if (t_1 <= 0.9972580999534978) {
		tmp = (fma(fma(fma((th * th), -0.0001984126984126984, 0.008333333333333333), (th * th), -0.16666666666666666), (th * th), 1.0) * th) * (sin(ky) / sqrt(fma((1.0 - t_2), 0.5, ((1.0 - cos((kx + kx))) * 0.5))));
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = cos(Float64(ky + ky))
	t_3 = Float64(Float64(sin(ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th))
	tmp = 0.0
	if (t_1 <= -0.945)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / Float64(Float64(2.0 - t_2) - cos(Float64(2.0 * kx)))))) * Float64(th * sin(ky)));
	elseif (t_1 <= 1e-7)
		tmp = t_3;
	elseif (t_1 <= 0.9972580999534978)
		tmp = Float64(Float64(fma(fma(fma(Float64(th * th), -0.0001984126984126984, 0.008333333333333333), Float64(th * th), -0.16666666666666666), Float64(th * th), 1.0) * th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - t_2), 0.5, Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5)))));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_3;
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.945], N[(N[(N[(1.0 / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 - t$95$2), $MachinePrecision] - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], t$95$3, If[LessEqual[t$95$1, 0.9972580999534978], N[(N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5 + N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], t$95$3]]]]]]]]

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))))) < -0.944999999999999951

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -0.944999999999999951 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.3

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 60.8%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 60.8%

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

   12. Applied rewrites 60.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999995e-8 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 90.9%

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

      1. lift-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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99725809995349779

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.1

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 35.4%

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

   if 0.99725809995349779 < (/.f64 (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

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

      1. lower-sin.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{\frac{1}{\left(2 - \cos \left(ky + ky\right)\right) - \cos \left(2 \cdot kx\right)}}\right) \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9972580999534978:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, -0.0001984126984126984, 0.008333333333333333\right), th \cdot th, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \cos \left(ky + ky\right)\\ \mathbf{if}\;t\_1 \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{\frac{1}{\left(2 - t\_2\right) - \cos \left(2 \cdot kx\right)}}\right) \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.9972580999534978:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, -0.0001984126984126984, 0.008333333333333333\right), th \cdot th, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - t\_2, 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (cos (+ ky ky))))
   (if (<= t_1 -0.945)
     (* (* (/ 1.0 (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5))) (sin ky)) (sin th))
     (if (<= t_1 -0.1)
       (*
        (* (sqrt 2.0) (sqrt (/ 1.0 (- (- 2.0 t_2) (cos (* 2.0 kx))))))
        (* th (sin ky)))
       (if (<= t_1 1e-23)
         (* (/ (sin ky) (sin kx)) (sin th))
         (if (<= t_1 0.9972580999534978)
           (*
            (*
             (fma
              (fma
               (fma (* th th) -0.0001984126984126984 0.008333333333333333)
               (* th th)
               -0.16666666666666666)
              (* th th)
              1.0)
             th)
            (/
             (sin ky)
             (sqrt (fma (- 1.0 t_2) 0.5 (* (- 1.0 (cos (+ kx kx))) 0.5)))))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = cos((ky + ky));
	double tmp;
	if (t_1 <= -0.945) {
		tmp = ((1.0 / sqrt(((1.0 - cos((2.0 * ky))) * 0.5))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = (sqrt(2.0) * sqrt((1.0 / ((2.0 - t_2) - cos((2.0 * kx)))))) * (th * sin(ky));
	} else if (t_1 <= 1e-23) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.9972580999534978) {
		tmp = (fma(fma(fma((th * th), -0.0001984126984126984, 0.008333333333333333), (th * th), -0.16666666666666666), (th * th), 1.0) * th) * (sin(ky) / sqrt(fma((1.0 - t_2), 0.5, ((1.0 - cos((kx + kx))) * 0.5))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = cos(Float64(ky + ky))
	tmp = 0.0
	if (t_1 <= -0.945)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / Float64(Float64(2.0 - t_2) - cos(Float64(2.0 * kx)))))) * Float64(th * sin(ky)));
	elseif (t_1 <= 1e-23)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.9972580999534978)
		tmp = Float64(Float64(fma(fma(fma(Float64(th * th), -0.0001984126984126984, 0.008333333333333333), Float64(th * th), -0.16666666666666666), Float64(th * th), 1.0) * th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - t_2), 0.5, Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5)))));
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.945], N[(N[(N[(1.0 / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 - t$95$2), $MachinePrecision] - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-23], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9972580999534978], N[(N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5 + N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $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))))) < -0.944999999999999951

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -0.944999999999999951 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.3

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 60.8%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 60.8%

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

   12. Applied rewrites 60.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999996e-24

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if 9.9999999999999996e-24 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99725809995349779

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.1

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 37.3%

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

   if 0.99725809995349779 < (/.f64 (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 82.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{\frac{1}{\left(2 - \cos \left(ky + ky\right)\right) - \cos \left(2 \cdot kx\right)}}\right) \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9972580999534978:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, -0.0001984126984126984, 0.008333333333333333\right), th \cdot th, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \cos \left(ky + ky\right)\\ \mathbf{if}\;t\_1 \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{\frac{1}{\left(2 - t\_2\right) - \cos \left(2 \cdot kx\right)}}\right) \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.9972580999534978:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - t\_2, 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (cos (+ ky ky))))
   (if (<= t_1 -0.945)
     (* (* (/ 1.0 (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5))) (sin ky)) (sin th))
     (if (<= t_1 -0.1)
       (*
        (* (sqrt 2.0) (sqrt (/ 1.0 (- (- 2.0 t_2) (cos (* 2.0 kx))))))
        (* th (sin ky)))
       (if (<= t_1 1e-23)
         (* (/ (sin ky) (sin kx)) (sin th))
         (if (<= t_1 0.9972580999534978)
           (*
            (*
             (fma
              (fma (* th th) 0.008333333333333333 -0.16666666666666666)
              (* th th)
              1.0)
             th)
            (/
             (sin ky)
             (sqrt (fma (- 1.0 t_2) 0.5 (* (- 1.0 (cos (+ kx kx))) 0.5)))))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = cos((ky + ky));
	double tmp;
	if (t_1 <= -0.945) {
		tmp = ((1.0 / sqrt(((1.0 - cos((2.0 * ky))) * 0.5))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = (sqrt(2.0) * sqrt((1.0 / ((2.0 - t_2) - cos((2.0 * kx)))))) * (th * sin(ky));
	} else if (t_1 <= 1e-23) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.9972580999534978) {
		tmp = (fma(fma((th * th), 0.008333333333333333, -0.16666666666666666), (th * th), 1.0) * th) * (sin(ky) / sqrt(fma((1.0 - t_2), 0.5, ((1.0 - cos((kx + kx))) * 0.5))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = cos(Float64(ky + ky))
	tmp = 0.0
	if (t_1 <= -0.945)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / Float64(Float64(2.0 - t_2) - cos(Float64(2.0 * kx)))))) * Float64(th * sin(ky)));
	elseif (t_1 <= 1e-23)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.9972580999534978)
		tmp = Float64(Float64(fma(fma(Float64(th * th), 0.008333333333333333, -0.16666666666666666), Float64(th * th), 1.0) * th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - t_2), 0.5, Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5)))));
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.945], N[(N[(N[(1.0 / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 - t$95$2), $MachinePrecision] - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-23], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9972580999534978], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5 + N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $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))))) < -0.944999999999999951

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -0.944999999999999951 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.3

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 60.8%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 60.8%

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

   12. Applied rewrites 60.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999996e-24

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if 9.9999999999999996e-24 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99725809995349779

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.1

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 37.4

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

   7. Applied rewrites 37.4%

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

   if 0.99725809995349779 < (/.f64 (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 82.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{\frac{1}{\left(2 - \cos \left(ky + ky\right)\right) - \cos \left(2 \cdot kx\right)}}\right) \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9972580999534978:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\right) \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \cos \left(2 \cdot ky\right)\\ t_3 := \cos \left(2 \cdot kx\right)\\ \mathbf{if}\;t\_1 \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - t\_2\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{\frac{1}{\left(2 - \cos \left(ky + ky\right)\right) - t\_3}}\right) \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.9972580999534978:\\ \;\;\;\;\left(\sqrt{\frac{1}{\left(\left(1 - t\_3\right) - \left(t\_2 - 1\right)\right) \cdot 0.5}} \cdot \left(-th\right)\right) \cdot \left(-\sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (cos (* 2.0 ky)))
        (t_3 (cos (* 2.0 kx))))
   (if (<= t_1 -0.945)
     (* (* (/ 1.0 (sqrt (* (- 1.0 t_2) 0.5))) (sin ky)) (sin th))
     (if (<= t_1 -0.1)
       (*
        (* (sqrt 2.0) (sqrt (/ 1.0 (- (- 2.0 (cos (+ ky ky))) t_3))))
        (* th (sin ky)))
       (if (<= t_1 1e-23)
         (* (/ (sin ky) (sin kx)) (sin th))
         (if (<= t_1 0.9972580999534978)
           (*
            (* (sqrt (/ 1.0 (* (- (- 1.0 t_3) (- t_2 1.0)) 0.5))) (- th))
            (- (sin ky)))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = cos((2.0 * ky));
	double t_3 = cos((2.0 * kx));
	double tmp;
	if (t_1 <= -0.945) {
		tmp = ((1.0 / sqrt(((1.0 - t_2) * 0.5))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = (sqrt(2.0) * sqrt((1.0 / ((2.0 - cos((ky + ky))) - t_3)))) * (th * sin(ky));
	} else if (t_1 <= 1e-23) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.9972580999534978) {
		tmp = (sqrt((1.0 / (((1.0 - t_3) - (t_2 - 1.0)) * 0.5))) * -th) * -sin(ky);
	} 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) :: t_3
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    t_2 = cos((2.0d0 * ky))
    t_3 = cos((2.0d0 * kx))
    if (t_1 <= (-0.945d0)) then
        tmp = ((1.0d0 / sqrt(((1.0d0 - t_2) * 0.5d0))) * sin(ky)) * sin(th)
    else if (t_1 <= (-0.1d0)) then
        tmp = (sqrt(2.0d0) * sqrt((1.0d0 / ((2.0d0 - cos((ky + ky))) - t_3)))) * (th * sin(ky))
    else if (t_1 <= 1d-23) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else if (t_1 <= 0.9972580999534978d0) then
        tmp = (sqrt((1.0d0 / (((1.0d0 - t_3) - (t_2 - 1.0d0)) * 0.5d0))) * -th) * -sin(ky)
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_2 = Math.cos((2.0 * ky));
	double t_3 = Math.cos((2.0 * kx));
	double tmp;
	if (t_1 <= -0.945) {
		tmp = ((1.0 / Math.sqrt(((1.0 - t_2) * 0.5))) * Math.sin(ky)) * Math.sin(th);
	} else if (t_1 <= -0.1) {
		tmp = (Math.sqrt(2.0) * Math.sqrt((1.0 / ((2.0 - Math.cos((ky + ky))) - t_3)))) * (th * Math.sin(ky));
	} else if (t_1 <= 1e-23) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else if (t_1 <= 0.9972580999534978) {
		tmp = (Math.sqrt((1.0 / (((1.0 - t_3) - (t_2 - 1.0)) * 0.5))) * -th) * -Math.sin(ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_2 = math.cos((2.0 * ky))
	t_3 = math.cos((2.0 * kx))
	tmp = 0
	if t_1 <= -0.945:
		tmp = ((1.0 / math.sqrt(((1.0 - t_2) * 0.5))) * math.sin(ky)) * math.sin(th)
	elif t_1 <= -0.1:
		tmp = (math.sqrt(2.0) * math.sqrt((1.0 / ((2.0 - math.cos((ky + ky))) - t_3)))) * (th * math.sin(ky))
	elif t_1 <= 1e-23:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	elif t_1 <= 0.9972580999534978:
		tmp = (math.sqrt((1.0 / (((1.0 - t_3) - (t_2 - 1.0)) * 0.5))) * -th) * -math.sin(ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = cos(Float64(2.0 * ky))
	t_3 = cos(Float64(2.0 * kx))
	tmp = 0.0
	if (t_1 <= -0.945)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(1.0 - t_2) * 0.5))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 / Float64(Float64(2.0 - cos(Float64(ky + ky))) - t_3)))) * Float64(th * sin(ky)));
	elseif (t_1 <= 1e-23)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.9972580999534978)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(Float64(Float64(1.0 - t_3) - Float64(t_2 - 1.0)) * 0.5))) * Float64(-th)) * Float64(-sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_2 = cos((2.0 * ky));
	t_3 = cos((2.0 * kx));
	tmp = 0.0;
	if (t_1 <= -0.945)
		tmp = ((1.0 / sqrt(((1.0 - t_2) * 0.5))) * sin(ky)) * sin(th);
	elseif (t_1 <= -0.1)
		tmp = (sqrt(2.0) * sqrt((1.0 / ((2.0 - cos((ky + ky))) - t_3)))) * (th * sin(ky));
	elseif (t_1 <= 1e-23)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	elseif (t_1 <= 0.9972580999534978)
		tmp = (sqrt((1.0 / (((1.0 - t_3) - (t_2 - 1.0)) * 0.5))) * -th) * -sin(ky);
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.945], N[(N[(N[(1.0 / N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-23], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9972580999534978], N[(N[(N[Sqrt[N[(1.0 / N[(N[(N[(1.0 - t$95$3), $MachinePrecision] - N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-th)), $MachinePrecision] * (-N[Sin[ky], $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))))) < -0.944999999999999951

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -0.944999999999999951 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.3

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 60.8%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 60.8%

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

   12. Applied rewrites 60.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999996e-24

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if 9.9999999999999996e-24 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99725809995349779

   1. Initial program 99.5%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 38.2%

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

   if 0.99725809995349779 < (/.f64 (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 82.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{\frac{1}{\left(2 - \cos \left(ky + ky\right)\right) - \cos \left(2 \cdot kx\right)}}\right) \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9972580999534978:\\ \;\;\;\;\left(\sqrt{\frac{1}{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) - \left(\cos \left(2 \cdot ky\right) - 1\right)\right) \cdot 0.5}} \cdot \left(-th\right)\right) \cdot \left(-\sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \cos \left(2 \cdot ky\right)\\ t_3 := \left(\sqrt{\frac{1}{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) - \left(t\_2 - 1\right)\right) \cdot 0.5}} \cdot \left(-th\right)\right) \cdot \left(-\sin ky\right)\\ \mathbf{if}\;t\_1 \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - t\_2\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.9972580999534978:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (cos (* 2.0 ky)))
        (t_3
         (*
          (*
           (sqrt (/ 1.0 (* (- (- 1.0 (cos (* 2.0 kx))) (- t_2 1.0)) 0.5)))
           (- th))
          (- (sin ky)))))
   (if (<= t_1 -0.945)
     (* (* (/ 1.0 (sqrt (* (- 1.0 t_2) 0.5))) (sin ky)) (sin th))
     (if (<= t_1 -0.1)
       t_3
       (if (<= t_1 1e-23)
         (* (/ (sin ky) (sin kx)) (sin th))
         (if (<= t_1 0.9972580999534978) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = cos((2.0 * ky));
	double t_3 = (sqrt((1.0 / (((1.0 - cos((2.0 * kx))) - (t_2 - 1.0)) * 0.5))) * -th) * -sin(ky);
	double tmp;
	if (t_1 <= -0.945) {
		tmp = ((1.0 / sqrt(((1.0 - t_2) * 0.5))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = t_3;
	} else if (t_1 <= 1e-23) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.9972580999534978) {
		tmp = t_3;
	} 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) :: t_3
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    t_2 = cos((2.0d0 * ky))
    t_3 = (sqrt((1.0d0 / (((1.0d0 - cos((2.0d0 * kx))) - (t_2 - 1.0d0)) * 0.5d0))) * -th) * -sin(ky)
    if (t_1 <= (-0.945d0)) then
        tmp = ((1.0d0 / sqrt(((1.0d0 - t_2) * 0.5d0))) * sin(ky)) * sin(th)
    else if (t_1 <= (-0.1d0)) then
        tmp = t_3
    else if (t_1 <= 1d-23) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else if (t_1 <= 0.9972580999534978d0) then
        tmp = t_3
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_2 = Math.cos((2.0 * ky));
	double t_3 = (Math.sqrt((1.0 / (((1.0 - Math.cos((2.0 * kx))) - (t_2 - 1.0)) * 0.5))) * -th) * -Math.sin(ky);
	double tmp;
	if (t_1 <= -0.945) {
		tmp = ((1.0 / Math.sqrt(((1.0 - t_2) * 0.5))) * Math.sin(ky)) * Math.sin(th);
	} else if (t_1 <= -0.1) {
		tmp = t_3;
	} else if (t_1 <= 1e-23) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else if (t_1 <= 0.9972580999534978) {
		tmp = t_3;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_2 = math.cos((2.0 * ky))
	t_3 = (math.sqrt((1.0 / (((1.0 - math.cos((2.0 * kx))) - (t_2 - 1.0)) * 0.5))) * -th) * -math.sin(ky)
	tmp = 0
	if t_1 <= -0.945:
		tmp = ((1.0 / math.sqrt(((1.0 - t_2) * 0.5))) * math.sin(ky)) * math.sin(th)
	elif t_1 <= -0.1:
		tmp = t_3
	elif t_1 <= 1e-23:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	elif t_1 <= 0.9972580999534978:
		tmp = t_3
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = cos(Float64(2.0 * ky))
	t_3 = Float64(Float64(sqrt(Float64(1.0 / Float64(Float64(Float64(1.0 - cos(Float64(2.0 * kx))) - Float64(t_2 - 1.0)) * 0.5))) * Float64(-th)) * Float64(-sin(ky)))
	tmp = 0.0
	if (t_1 <= -0.945)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(1.0 - t_2) * 0.5))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = t_3;
	elseif (t_1 <= 1e-23)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.9972580999534978)
		tmp = t_3;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_2 = cos((2.0 * ky));
	t_3 = (sqrt((1.0 / (((1.0 - cos((2.0 * kx))) - (t_2 - 1.0)) * 0.5))) * -th) * -sin(ky);
	tmp = 0.0;
	if (t_1 <= -0.945)
		tmp = ((1.0 / sqrt(((1.0 - t_2) * 0.5))) * sin(ky)) * sin(th);
	elseif (t_1 <= -0.1)
		tmp = t_3;
	elseif (t_1 <= 1e-23)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	elseif (t_1 <= 0.9972580999534978)
		tmp = t_3;
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[N[(1.0 / N[(N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-th)), $MachinePrecision] * (-N[Sin[ky], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[t$95$1, -0.945], N[(N[(N[(1.0 / N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], t$95$3, If[LessEqual[t$95$1, 1e-23], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9972580999534978], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.944999999999999951

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -0.944999999999999951 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 9.9999999999999996e-24 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99725809995349779

   1. Initial program 99.4%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 49.3%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999996e-24

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if 0.99725809995349779 < (/.f64 (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 82.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\sqrt{\frac{1}{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) - \left(\cos \left(2 \cdot ky\right) - 1\right)\right) \cdot 0.5}} \cdot \left(-th\right)\right) \cdot \left(-\sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9972580999534978:\\ \;\;\;\;\left(\sqrt{\frac{1}{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) - \left(\cos \left(2 \cdot ky\right) - 1\right)\right) \cdot 0.5}} \cdot \left(-th\right)\right) \cdot \left(-\sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(2 \cdot kx\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_3 := \cos \left(2 \cdot ky\right)\\ t_4 := 1 - t\_3\\ t_5 := th \cdot \sin ky\\ \mathbf{if}\;t\_2 \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{t\_4 \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\sqrt{\frac{2}{1 - \left(t\_1 - t\_4\right)}} \cdot t\_5\\ \mathbf{elif}\;t\_2 \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.9972580999534978:\\ \;\;\;\;\sqrt{\frac{2}{\left(2 - t\_3\right) - t\_1}} \cdot t\_5\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* 2.0 kx)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_3 (cos (* 2.0 ky)))
        (t_4 (- 1.0 t_3))
        (t_5 (* th (sin ky))))
   (if (<= t_2 -0.945)
     (* (* (/ 1.0 (sqrt (* t_4 0.5))) (sin ky)) (sin th))
     (if (<= t_2 -0.1)
       (* (sqrt (/ 2.0 (- 1.0 (- t_1 t_4)))) t_5)
       (if (<= t_2 1e-23)
         (* (/ (sin ky) (sin kx)) (sin th))
         (if (<= t_2 0.9972580999534978)
           (* (sqrt (/ 2.0 (- (- 2.0 t_3) t_1))) t_5)
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((2.0 * kx));
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_3 = cos((2.0 * ky));
	double t_4 = 1.0 - t_3;
	double t_5 = th * sin(ky);
	double tmp;
	if (t_2 <= -0.945) {
		tmp = ((1.0 / sqrt((t_4 * 0.5))) * sin(ky)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = sqrt((2.0 / (1.0 - (t_1 - t_4)))) * t_5;
	} else if (t_2 <= 1e-23) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_2 <= 0.9972580999534978) {
		tmp = sqrt((2.0 / ((2.0 - t_3) - t_1))) * t_5;
	} 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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = cos((2.0d0 * kx))
    t_2 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    t_3 = cos((2.0d0 * ky))
    t_4 = 1.0d0 - t_3
    t_5 = th * sin(ky)
    if (t_2 <= (-0.945d0)) then
        tmp = ((1.0d0 / sqrt((t_4 * 0.5d0))) * sin(ky)) * sin(th)
    else if (t_2 <= (-0.1d0)) then
        tmp = sqrt((2.0d0 / (1.0d0 - (t_1 - t_4)))) * t_5
    else if (t_2 <= 1d-23) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else if (t_2 <= 0.9972580999534978d0) then
        tmp = sqrt((2.0d0 / ((2.0d0 - t_3) - t_1))) * t_5
    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.cos((2.0 * kx));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_3 = Math.cos((2.0 * ky));
	double t_4 = 1.0 - t_3;
	double t_5 = th * Math.sin(ky);
	double tmp;
	if (t_2 <= -0.945) {
		tmp = ((1.0 / Math.sqrt((t_4 * 0.5))) * Math.sin(ky)) * Math.sin(th);
	} else if (t_2 <= -0.1) {
		tmp = Math.sqrt((2.0 / (1.0 - (t_1 - t_4)))) * t_5;
	} else if (t_2 <= 1e-23) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else if (t_2 <= 0.9972580999534978) {
		tmp = Math.sqrt((2.0 / ((2.0 - t_3) - t_1))) * t_5;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.cos((2.0 * kx))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_3 = math.cos((2.0 * ky))
	t_4 = 1.0 - t_3
	t_5 = th * math.sin(ky)
	tmp = 0
	if t_2 <= -0.945:
		tmp = ((1.0 / math.sqrt((t_4 * 0.5))) * math.sin(ky)) * math.sin(th)
	elif t_2 <= -0.1:
		tmp = math.sqrt((2.0 / (1.0 - (t_1 - t_4)))) * t_5
	elif t_2 <= 1e-23:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	elif t_2 <= 0.9972580999534978:
		tmp = math.sqrt((2.0 / ((2.0 - t_3) - t_1))) * t_5
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(2.0 * kx))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_3 = cos(Float64(2.0 * ky))
	t_4 = Float64(1.0 - t_3)
	t_5 = Float64(th * sin(ky))
	tmp = 0.0
	if (t_2 <= -0.945)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(t_4 * 0.5))) * sin(ky)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(sqrt(Float64(2.0 / Float64(1.0 - Float64(t_1 - t_4)))) * t_5);
	elseif (t_2 <= 1e-23)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_2 <= 0.9972580999534978)
		tmp = Float64(sqrt(Float64(2.0 / Float64(Float64(2.0 - t_3) - t_1))) * t_5);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = cos((2.0 * kx));
	t_2 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_3 = cos((2.0 * ky));
	t_4 = 1.0 - t_3;
	t_5 = th * sin(ky);
	tmp = 0.0;
	if (t_2 <= -0.945)
		tmp = ((1.0 / sqrt((t_4 * 0.5))) * sin(ky)) * sin(th);
	elseif (t_2 <= -0.1)
		tmp = sqrt((2.0 / (1.0 - (t_1 - t_4)))) * t_5;
	elseif (t_2 <= 1e-23)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	elseif (t_2 <= 0.9972580999534978)
		tmp = sqrt((2.0 / ((2.0 - t_3) - t_1))) * t_5;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.945], N[(N[(N[(1.0 / N[Sqrt[N[(t$95$4 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[Sqrt[N[(2.0 / N[(1.0 - N[(t$95$1 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision], If[LessEqual[t$95$2, 1e-23], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9972580999534978], N[(N[Sqrt[N[(2.0 / N[(N[(2.0 - t$95$3), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$5), $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))))) < -0.944999999999999951

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -0.944999999999999951 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.3

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 60.8%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

   10. Applied rewrites 60.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999996e-24

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if 9.9999999999999996e-24 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99725809995349779

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.1

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 37.9%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 38.0%

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

   if 0.99725809995349779 < (/.f64 (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 82.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\sqrt{\frac{2}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9972580999534978:\\ \;\;\;\;\sqrt{\frac{2}{\left(2 - \cos \left(2 \cdot ky\right)\right) - \cos \left(2 \cdot kx\right)}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \cos \left(2 \cdot ky\right)\\ t_3 := \sqrt{\frac{2}{\left(2 - t\_2\right) - \cos \left(2 \cdot kx\right)}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{if}\;t\_1 \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - t\_2\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.9972580999534978:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (cos (* 2.0 ky)))
        (t_3
         (* (sqrt (/ 2.0 (- (- 2.0 t_2) (cos (* 2.0 kx))))) (* th (sin ky)))))
   (if (<= t_1 -0.945)
     (* (* (/ 1.0 (sqrt (* (- 1.0 t_2) 0.5))) (sin ky)) (sin th))
     (if (<= t_1 -0.1)
       t_3
       (if (<= t_1 1e-23)
         (* (/ (sin ky) (sin kx)) (sin th))
         (if (<= t_1 0.9972580999534978) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = cos((2.0 * ky));
	double t_3 = sqrt((2.0 / ((2.0 - t_2) - cos((2.0 * kx))))) * (th * sin(ky));
	double tmp;
	if (t_1 <= -0.945) {
		tmp = ((1.0 / sqrt(((1.0 - t_2) * 0.5))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = t_3;
	} else if (t_1 <= 1e-23) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.9972580999534978) {
		tmp = t_3;
	} 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) :: t_3
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    t_2 = cos((2.0d0 * ky))
    t_3 = sqrt((2.0d0 / ((2.0d0 - t_2) - cos((2.0d0 * kx))))) * (th * sin(ky))
    if (t_1 <= (-0.945d0)) then
        tmp = ((1.0d0 / sqrt(((1.0d0 - t_2) * 0.5d0))) * sin(ky)) * sin(th)
    else if (t_1 <= (-0.1d0)) then
        tmp = t_3
    else if (t_1 <= 1d-23) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else if (t_1 <= 0.9972580999534978d0) then
        tmp = t_3
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_2 = Math.cos((2.0 * ky));
	double t_3 = Math.sqrt((2.0 / ((2.0 - t_2) - Math.cos((2.0 * kx))))) * (th * Math.sin(ky));
	double tmp;
	if (t_1 <= -0.945) {
		tmp = ((1.0 / Math.sqrt(((1.0 - t_2) * 0.5))) * Math.sin(ky)) * Math.sin(th);
	} else if (t_1 <= -0.1) {
		tmp = t_3;
	} else if (t_1 <= 1e-23) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else if (t_1 <= 0.9972580999534978) {
		tmp = t_3;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_2 = math.cos((2.0 * ky))
	t_3 = math.sqrt((2.0 / ((2.0 - t_2) - math.cos((2.0 * kx))))) * (th * math.sin(ky))
	tmp = 0
	if t_1 <= -0.945:
		tmp = ((1.0 / math.sqrt(((1.0 - t_2) * 0.5))) * math.sin(ky)) * math.sin(th)
	elif t_1 <= -0.1:
		tmp = t_3
	elif t_1 <= 1e-23:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	elif t_1 <= 0.9972580999534978:
		tmp = t_3
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = cos(Float64(2.0 * ky))
	t_3 = Float64(sqrt(Float64(2.0 / Float64(Float64(2.0 - t_2) - cos(Float64(2.0 * kx))))) * Float64(th * sin(ky)))
	tmp = 0.0
	if (t_1 <= -0.945)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(1.0 - t_2) * 0.5))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = t_3;
	elseif (t_1 <= 1e-23)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.9972580999534978)
		tmp = t_3;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_2 = cos((2.0 * ky));
	t_3 = sqrt((2.0 / ((2.0 - t_2) - cos((2.0 * kx))))) * (th * sin(ky));
	tmp = 0.0;
	if (t_1 <= -0.945)
		tmp = ((1.0 / sqrt(((1.0 - t_2) * 0.5))) * sin(ky)) * sin(th);
	elseif (t_1 <= -0.1)
		tmp = t_3;
	elseif (t_1 <= 1e-23)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	elseif (t_1 <= 0.9972580999534978)
		tmp = t_3;
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(2.0 / N[(N[(2.0 - t$95$2), $MachinePrecision] - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.945], N[(N[(N[(1.0 / N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], t$95$3, If[LessEqual[t$95$1, 1e-23], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9972580999534978], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.944999999999999951

   1. Initial program 90.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if -0.944999999999999951 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 9.9999999999999996e-24 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99725809995349779

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.2

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 49.2%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999996e-24

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if 0.99725809995349779 < (/.f64 (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 82.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.945:\\ \;\;\;\;\left(\frac{1}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\sqrt{\frac{2}{\left(2 - \cos \left(2 \cdot ky\right)\right) - \cos \left(2 \cdot kx\right)}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9972580999534978:\\ \;\;\;\;\sqrt{\frac{2}{\left(2 - \cos \left(2 \cdot ky\right)\right) - \cos \left(2 \cdot kx\right)}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \frac{\sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -1.0)
     (*
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/ (sin ky) (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5))))
     (if (<= t_1 2e-257)
       (* (/ (sin th) (sqrt (* (- 1.0 (cos (+ kx kx))) 0.5))) (sin ky))
       (if (<= t_1 1e-7) (* (/ ky (sin kx)) (sin th)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) * (sin(ky) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5)));
	} else if (t_1 <= 2e-257) {
		tmp = (sin(th) / sqrt(((1.0 - cos((kx + kx))) * 0.5))) * sin(ky);
	} else if (t_1 <= 1e-7) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * Float64(sin(ky) / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5))));
	elseif (t_1 <= 2e-257)
		tmp = Float64(Float64(sin(th) / sqrt(Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5))) * sin(ky));
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-257], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $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))))) < -1

   1. Initial program 89.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 68.9

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.3

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 33.0

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

   10. Applied rewrites 33.0%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-257

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 99.5

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 10.1

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

   7. Applied rewrites 10.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 10.1

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

   9. Applied rewrites 52.2%

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

   if 2e-257 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999995e-8

   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

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 60.6

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

   5. Applied rewrites 60.6%

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

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 63.9

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

   5. Applied rewrites 63.9%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \frac{\sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2 \cdot 10^{-257}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.1)
     (* (/ (sin ky) (sqrt (* 0.5 (- 1.0 (cos (+ ky ky)))))) (sin th))
     (if (<= t_1 1e-7) (* (/ (sin ky) (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (sin(ky) / sqrt((0.5 * (1.0 - cos((ky + ky)))))) * sin(th);
	} else if (t_1 <= 1e-7) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.1d0)) then
        tmp = (sin(ky) / sqrt((0.5d0 * (1.0d0 - cos((ky + ky)))))) * sin(th)
    else if (t_1 <= 1d-7) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 * (1.0 - Math.cos((ky + ky)))))) * Math.sin(th);
	} else if (t_1 <= 1e-7) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.1:
		tmp = (math.sin(ky) / math.sqrt((0.5 * (1.0 - math.cos((ky + ky)))))) * math.sin(th)
	elif t_1 <= 1e-7:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(ky + ky)))))) * sin(th));
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = (sin(ky) / sqrt((0.5 * (1.0 - cos((ky + ky)))))) * sin(th);
	elseif (t_1 <= 1e-7)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 83.4

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 46.2

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

   7. Applied rewrites 46.2%

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

   9. Applied rewrites 46.2%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999995e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 63.9

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

   5. Applied rewrites 63.9%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -1.0)
     (*
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/ (sin ky) (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5))))
     (if (<= t_1 1e-7) (* (/ (sin ky) (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) * (sin(ky) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5)));
	} else if (t_1 <= 1e-7) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * Float64(sin(ky) / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5))));
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 89.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. count-2 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. count-2 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-+.f64 68.9

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lift-sin.f64 N/A

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

      21. lift-sin.f64 N/A

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

      22. sin-mult N/A

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.3

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th\right) \]

      7. lower-*.f64 33.0

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th\right) \]

   10. Applied rewrites 33.0%

       \[\leadsto \frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999995e-8

   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 48.2

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 48.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \frac{\sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\sqrt{\frac{-1}{\cos \left(2 \cdot ky\right) - 1}} \cdot \left(\sqrt{2} \cdot \left(th \cdot \sin ky\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.1)
     (*
      (sqrt (/ -1.0 (- (cos (* 2.0 ky)) 1.0)))
      (* (sqrt 2.0) (* th (sin ky))))
     (if (<= t_1 1e-7) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = sqrt((-1.0 / (cos((2.0 * ky)) - 1.0))) * (sqrt(2.0) * (th * sin(ky)));
	} else if (t_1 <= 1e-7) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.1d0)) then
        tmp = sqrt(((-1.0d0) / (cos((2.0d0 * ky)) - 1.0d0))) * (sqrt(2.0d0) * (th * sin(ky)))
    else if (t_1 <= 1d-7) then
        tmp = (ky / sin(kx)) * sin(th)
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = Math.sqrt((-1.0 / (Math.cos((2.0 * ky)) - 1.0))) * (Math.sqrt(2.0) * (th * Math.sin(ky)));
	} else if (t_1 <= 1e-7) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.1:
		tmp = math.sqrt((-1.0 / (math.cos((2.0 * ky)) - 1.0))) * (math.sqrt(2.0) * (th * math.sin(ky)))
	elif t_1 <= 1e-7:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(sqrt(Float64(-1.0 / Float64(cos(Float64(2.0 * ky)) - 1.0))) * Float64(sqrt(2.0) * Float64(th * sin(ky))));
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = sqrt((-1.0 / (cos((2.0 * ky)) - 1.0))) * (sqrt(2.0) * (th * sin(ky)));
	elseif (t_1 <= 1e-7)
		tmp = (ky / sin(kx)) * sin(th);
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[Sqrt[N[(-1.0 / N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      8. div-inv N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      11. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower-+.f64 83.4

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      19. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      20. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      21. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      22. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}\right)}} \cdot \sin th \]

   4. Applied rewrites 83.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      7. distribute-lft-out N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{2} \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{2} \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \color{blue}{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{0.5 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(ky \cdot 2\right)\right)\right)}}} \]

   8. Taylor expanded in kx around inf

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{1}{2 - \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)}}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 43.0%

       \[\leadsto \left(\sin ky \cdot th\right) \cdot \left(\sqrt{\frac{1}{\left(2 - \cos \left(ky \cdot 2\right)\right) - \cos \left(kx \cdot 2\right)}} \cdot \color{blue}{\sqrt{2}}\right) \]

   11. Taylor expanded in kx around 0

       \[\leadsto \left(th \cdot \left(\sin ky \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 23.5%

       \[\leadsto \left(\left(th \cdot \sin ky\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \]

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999995e-8

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in 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 63.5

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\sqrt{\frac{-1}{\cos \left(2 \cdot ky\right) - 1}} \cdot \left(\sqrt{2} \cdot \left(th \cdot \sin ky\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\sqrt{\frac{2}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.1)
     (* (sqrt (/ 2.0 (- 1.0 (cos (* 2.0 ky))))) (* th (sin ky)))
     (if (<= t_1 1e-7) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = sqrt((2.0 / (1.0 - cos((2.0 * ky))))) * (th * sin(ky));
	} else if (t_1 <= 1e-7) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.1d0)) then
        tmp = sqrt((2.0d0 / (1.0d0 - cos((2.0d0 * ky))))) * (th * sin(ky))
    else if (t_1 <= 1d-7) then
        tmp = (ky / sin(kx)) * sin(th)
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = Math.sqrt((2.0 / (1.0 - Math.cos((2.0 * ky))))) * (th * Math.sin(ky));
	} else if (t_1 <= 1e-7) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.1:
		tmp = math.sqrt((2.0 / (1.0 - math.cos((2.0 * ky))))) * (th * math.sin(ky))
	elif t_1 <= 1e-7:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(2.0 * ky))))) * Float64(th * sin(ky)));
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = sqrt((2.0 / (1.0 - cos((2.0 * ky))))) * (th * sin(ky));
	elseif (t_1 <= 1e-7)
		tmp = (ky / sin(kx)) * sin(th);
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-7], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      8. div-inv N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      11. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower-+.f64 83.4

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      19. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      20. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      21. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      22. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}\right)}} \cdot \sin th \]

   4. Applied rewrites 83.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      7. distribute-lft-out N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{2} \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{2} \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \color{blue}{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{0.5 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(ky \cdot 2\right)\right)\right)}}} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(2 \cdot ky\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 23.5%

       \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \cos \left(ky \cdot 2\right)}} \]

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999995e-8

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in 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 63.5

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\sqrt{\frac{2}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-7)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-7) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-7) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-7) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-7:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-7)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-7)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $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))))) < 9.9999999999999995e-8

   1. Initial program 96.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 35.1

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-7)
   (* (/ (sin th) (sin kx)) ky)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-7) {
		tmp = (sin(th) / sin(kx)) * ky;
	} 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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-7) then
        tmp = (sin(th) / sin(kx)) * ky
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-7) {
		tmp = (Math.sin(th) / Math.sin(kx)) * ky;
	} 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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-7:
		tmp = (math.sin(th) / math.sin(kx)) * ky
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-7)
		tmp = Float64(Float64(sin(th) / sin(kx)) * ky);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-7)
		tmp = (sin(th) / sin(kx)) * ky;
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * ky), $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))))) < 9.9999999999999995e-8

   1. Initial program 96.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right) \cdot ky} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right) \cdot ky} \]

   5. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{\sin th}{{\sin kx}^{3}}, -0.5, \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \frac{\sin th}{\sin kx}\right) \cdot ky} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\sin kx} \cdot ky \]
   7. Step-by-step derivation

   8. Applied rewrites 35.1%

      \[\leadsto \frac{\sin th}{\sin kx} \cdot ky \]

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 34.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-7)
   (/ (* (sin th) ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-7) {
		tmp = (sin(th) * ky) / (fma(-0.16666666666666666, (kx * kx), 1.0) * kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-7)
		tmp = Float64(Float64(sin(th) * ky) / Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * 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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * 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))))) < 9.9999999999999995e-8

   1. Initial program 96.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   7. 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 34.6

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sin kx}} \]

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{ky \cdot \sin th}{kx \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.7%

       \[\leadsto \frac{ky \cdot \sin th}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot \color{blue}{kx}} \]

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 14.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sin th \leq 10^{-323}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;1 \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))
       (sin th))
      1e-323)
   (* (* (* th th) th) -0.16666666666666666)
   (* 1.0 th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) * sin(th)) <= 1e-323) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = 1.0 * 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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) * sin(th)) <= 1d-323) then
        tmp = ((th * th) * th) * (-0.16666666666666666d0)
    else
        tmp = 1.0d0 * 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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) * Math.sin(th)) <= 1e-323) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = 1.0 * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) * math.sin(th)) <= 1e-323:
		tmp = ((th * th) * th) * -0.16666666666666666
	else:
		tmp = 1.0 * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) * sin(th)) <= 1e-323)
		tmp = Float64(Float64(Float64(th * th) * th) * -0.16666666666666666);
	else
		tmp = Float64(1.0 * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) * sin(th)) <= 1e-323)
		tmp = ((th * th) * th) * -0.16666666666666666;
	else
		tmp = 1.0 * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 1e-323], N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(1.0 * 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)) < 9.88131e-324

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 20.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 20.8%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.9%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   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%

       \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   if 9.88131e-324 < (*.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 95.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 26.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 26.6%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.5%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around 0

      \[\leadsto 1 \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 15.4%

       \[\leadsto 1 \cdot th \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sin th \leq 10^{-323}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;1 \cdot th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 34.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th \cdot ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-7)
   (/ (* (sin th) ky) kx)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-7) {
		tmp = (sin(th) * ky) / kx;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-7) then
        tmp = (sin(th) * ky) / kx
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-7) {
		tmp = (Math.sin(th) * 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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-7:
		tmp = (math.sin(th) * 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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-7)
		tmp = Float64(Float64(sin(th) * 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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-7)
		tmp = (sin(th) * 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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / kx), $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))))) < 9.9999999999999995e-8

   1. Initial program 96.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   7. 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 34.6

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sin kx}} \]

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
   10. Step-by-step derivation

   11. Applied rewrites 24.1%

       \[\leadsto \frac{\sin th \cdot ky}{\color{blue}{kx}} \]

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{\sin th \cdot ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-7)
   (* (/ ky (sin kx)) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-7) {
		tmp = (ky / sin(kx)) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-7) then
        tmp = (ky / sin(kx)) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-7) {
		tmp = (ky / Math.sin(kx)) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-7:
		tmp = (ky / math.sin(kx)) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-7)
		tmp = Float64(Float64(ky / sin(kx)) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-7)
		tmp = (ky / sin(kx)) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * th), $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))))) < 9.9999999999999995e-8

   1. Initial program 96.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   7. 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 34.6

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sin kx}} \]

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

   9. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
   10. Step-by-step derivation

   11. Applied rewrites 24.2%

       \[\leadsto \frac{ky}{\sin kx} \cdot \color{blue}{th} \]

   if 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 30.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 9.5 \cdot 10^{-68}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 9.5e-68)
   (* (* (* th th) th) -0.16666666666666666)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 9.5e-68) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} 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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 9.5d-68) then
        tmp = ((th * th) * th) * (-0.16666666666666666d0)
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 9.5e-68) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} 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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 9.5e-68:
		tmp = ((th * th) * th) * -0.16666666666666666
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 9.5e-68)
		tmp = Float64(Float64(Float64(th * th) * th) * -0.16666666666666666);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 9.5e-68)
		tmp = ((th * th) * th) * -0.16666666666666666;
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 9.5e-68], N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666), $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))))) < 9.4999999999999997e-68

   1. Initial program 96.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.4%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 18.2%

       \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   if 9.4999999999999997e-68 < (/.f64 (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 90.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 58.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 9.5 \cdot 10^{-68}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 75.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.003:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.003)
   (*
    (/
     (sin ky)
     (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
    (sin th))
   (*
    (/
     (sin ky)
     (sqrt (fma (- 1.0 (cos (+ ky ky))) 0.5 (* (- 1.0 (cos (+ kx kx))) 0.5))))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.003) {
		tmp = (sin(ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, ((1.0 - cos((kx + kx))) * 0.5)))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.003)
		tmp = Float64(Float64(sin(ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5)))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.003], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0030000000000000001

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 63.1

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 63.1%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   if 0.0030000000000000001 < ky

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      8. div-inv N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      11. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower-+.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      19. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      20. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      21. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      22. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}\right)}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 13.0% accurate, 105.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* 1.0 th))

C

double code(double kx, double ky, double th) {
	return 1.0 * 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 = 1.0d0 * th
end function

Java

public static double code(double kx, double ky, double th) {
	return 1.0 * th;
}

Python

def code(kx, ky, th):
	return 1.0 * th

Julia

function code(kx, ky, th)
	return Float64(1.0 * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 1.0 * th;
end

Wolfram

code[kx_, ky_, th_] := N[(1.0 * th), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 23.0

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 23.0%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 12.0%

   \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

9. Taylor expanded in th around 0

   \[\leadsto 1 \cdot th \]
10. Step-by-step derivation

11. Applied rewrites 12.2%

    \[\leadsto 1 \cdot th \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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)))