Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.9% → 99.7%

Time: 15.6s

Alternatives: 31

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 2: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ t_2 := \cos \left(kx + kx\right)\\ t_3 := \cos \left(ky + ky\right)\\ t_4 := {\sin kx}^{2}\\ t_5 := \frac{\sin ky}{\sqrt{t\_4 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_5 \leq -0.98:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq -0.1:\\ \;\;\;\;\frac{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)}{\frac{\sqrt{\mathsf{fma}\left(1 - t\_2, 0.5, 0.5 + -0.5 \cdot t\_3\right)}}{\sin ky}}\\ \mathbf{elif}\;t\_5 \leq 0.2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_4 + ky \cdot ky}}\\ \mathbf{elif}\;t\_5 \leq 0.9991156854772065:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - t\_3, 0.5, \mathsf{fma}\left(t\_2, -0.5, 0.5\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ (sin ky) (hypot (sin ky) kx))))
        (t_2 (cos (+ kx kx)))
        (t_3 (cos (+ ky ky)))
        (t_4 (pow (sin kx) 2.0))
        (t_5 (/ (sin ky) (sqrt (+ t_4 (pow (sin ky) 2.0))))))
   (if (<= t_5 -0.98)
     t_1
     (if (<= t_5 -0.1)
       (/
        (* th (fma -0.16666666666666666 (* th th) 1.0))
        (/ (sqrt (fma (- 1.0 t_2) 0.5 (+ 0.5 (* -0.5 t_3)))) (sin ky)))
       (if (<= t_5 0.2)
         (* (sin th) (/ (sin ky) (sqrt (+ t_4 (* ky ky)))))
         (if (<= t_5 0.9991156854772065)
           (*
            (/ (sin ky) (sqrt (fma (- 1.0 t_3) 0.5 (fma t_2 -0.5 0.5))))
            (fma th (* -0.16666666666666666 (* th th)) th))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	double t_2 = cos((kx + kx));
	double t_3 = cos((ky + ky));
	double t_4 = pow(sin(kx), 2.0);
	double t_5 = sin(ky) / sqrt((t_4 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_5 <= -0.98) {
		tmp = t_1;
	} else if (t_5 <= -0.1) {
		tmp = (th * fma(-0.16666666666666666, (th * th), 1.0)) / (sqrt(fma((1.0 - t_2), 0.5, (0.5 + (-0.5 * t_3)))) / sin(ky));
	} else if (t_5 <= 0.2) {
		tmp = sin(th) * (sin(ky) / sqrt((t_4 + (ky * ky))));
	} else if (t_5 <= 0.9991156854772065) {
		tmp = (sin(ky) / sqrt(fma((1.0 - t_3), 0.5, fma(t_2, -0.5, 0.5)))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)))
	t_2 = cos(Float64(kx + kx))
	t_3 = cos(Float64(ky + ky))
	t_4 = sin(kx) ^ 2.0
	t_5 = Float64(sin(ky) / sqrt(Float64(t_4 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_5 <= -0.98)
		tmp = t_1;
	elseif (t_5 <= -0.1)
		tmp = Float64(Float64(th * fma(-0.16666666666666666, Float64(th * th), 1.0)) / Float64(sqrt(fma(Float64(1.0 - t_2), 0.5, Float64(0.5 + Float64(-0.5 * t_3)))) / sin(ky)));
	elseif (t_5 <= 0.2)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(t_4 + Float64(ky * ky)))));
	elseif (t_5 <= 0.9991156854772065)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(Float64(1.0 - t_3), 0.5, fma(t_2, -0.5, 0.5)))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$4 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -0.98], t$95$1, If[LessEqual[t$95$5, -0.1], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.2], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$4 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.9991156854772065], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$3), $MachinePrecision] * 0.5 + N[(t$95$2 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998 or 0.99911568547720653 < (/.f64 (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. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.9

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

   5. Simplified 89.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-hypot.f64 99.0

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

   7. Applied egg-rr 99.0%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 49.5

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

   7. Simplified 49.5%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 95.6

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

   5. Simplified 95.6%

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

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99911568547720653

   1. Initial program 99.2%

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

   4. Applied egg-rr 99.1%

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

   6. Applied egg-rr 23.2%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-fma.f64 23.2

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. lift-+.f64 99.2

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in th around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 40.7

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

   11. Simplified 40.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9991156854772065:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \left(kx + kx\right), -0.5, 0.5\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ t_2 := \cos \left(kx + kx\right)\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_4 := \cos \left(ky + ky\right)\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)}{\frac{\sqrt{\mathsf{fma}\left(1 - t\_2, 0.5, 0.5 + -0.5 \cdot t\_4\right)}}{\sin ky}}\\ \mathbf{elif}\;t\_3 \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;t\_3 \leq 0.9991156854772065:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - t\_4, 0.5, \mathsf{fma}\left(t\_2, -0.5, 0.5\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ (sin ky) (hypot (sin ky) kx))))
        (t_2 (cos (+ kx kx)))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_4 (cos (+ ky ky))))
   (if (<= t_3 -0.98)
     t_1
     (if (<= t_3 -0.1)
       (/
        (* th (fma -0.16666666666666666 (* th th) 1.0))
        (/ (sqrt (fma (- 1.0 t_2) 0.5 (+ 0.5 (* -0.5 t_4)))) (sin ky)))
       (if (<= t_3 0.001)
         (*
          (sin th)
          (/
           (* ky (fma (* ky ky) -0.16666666666666666 1.0))
           (hypot (sin ky) (sin kx))))
         (if (<= t_3 0.9991156854772065)
           (*
            (/ (sin ky) (sqrt (fma (- 1.0 t_4) 0.5 (fma t_2 -0.5 0.5))))
            (fma th (* -0.16666666666666666 (* th th)) th))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	double t_2 = cos((kx + kx));
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_4 = cos((ky + ky));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = t_1;
	} else if (t_3 <= -0.1) {
		tmp = (th * fma(-0.16666666666666666, (th * th), 1.0)) / (sqrt(fma((1.0 - t_2), 0.5, (0.5 + (-0.5 * t_4)))) / sin(ky));
	} else if (t_3 <= 0.001) {
		tmp = sin(th) * ((ky * fma((ky * ky), -0.16666666666666666, 1.0)) / hypot(sin(ky), sin(kx)));
	} else if (t_3 <= 0.9991156854772065) {
		tmp = (sin(ky) / sqrt(fma((1.0 - t_4), 0.5, fma(t_2, -0.5, 0.5)))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)))
	t_2 = cos(Float64(kx + kx))
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_4 = cos(Float64(ky + ky))
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = t_1;
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(th * fma(-0.16666666666666666, Float64(th * th), 1.0)) / Float64(sqrt(fma(Float64(1.0 - t_2), 0.5, Float64(0.5 + Float64(-0.5 * t_4)))) / sin(ky)));
	elseif (t_3 <= 0.001)
		tmp = Float64(sin(th) * Float64(Float64(ky * fma(Float64(ky * ky), -0.16666666666666666, 1.0)) / hypot(sin(ky), sin(kx))));
	elseif (t_3 <= 0.9991156854772065)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(Float64(1.0 - t_4), 0.5, fma(t_2, -0.5, 0.5)))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], t$95$1, If[LessEqual[t$95$3, -0.1], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.001], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9991156854772065], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$4), $MachinePrecision] * 0.5 + N[(t$95$2 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998 or 0.99911568547720653 < (/.f64 (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. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.9

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

   5. Simplified 89.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-hypot.f64 99.0

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

   7. Applied egg-rr 99.0%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 49.5

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

   7. Simplified 49.5%

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

   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))))) < 1e-3

   1. Initial program 99.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.5

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

   7. Simplified 99.5%

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

   if 1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99911568547720653

   1. Initial program 99.2%

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

   4. Applied egg-rr 99.1%

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

   6. Applied egg-rr 22.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-fma.f64 22.4

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. lift-+.f64 99.3

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in th around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 38.8

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

   11. Simplified 38.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{ky \cdot \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9991156854772065:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \left(kx + kx\right), -0.5, 0.5\right)\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(kx, kx, t\_1\right)}}\\ \mathbf{elif}\;t\_2 \leq 10^{-262}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -1.0)
     (* (* (sin ky) th) (sqrt (/ 1.0 (fma kx kx t_1))))
     (if (<= t_2 1e-262)
       (* (sin th) (* (sin ky) (sqrt (/ 2.0 (- 1.0 (cos (* kx -2.0)))))))
       (if (<= t_2 5e-60)
         (/ (* ky (sin th)) (sin kx))
         (if (<= t_2 0.0004)
           (*
            (sin th)
            (*
             (sqrt (/ 1.0 (fma (- 1.0 (cos (+ kx kx))) 0.5 (* ky ky))))
             (fma
              ky
              (*
               (* ky ky)
               (fma 0.008333333333333333 (* ky ky) -0.16666666666666666))
              ky)))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(kx, kx, t_1)));
	} else if (t_2 <= 1e-262) {
		tmp = sin(th) * (sin(ky) * sqrt((2.0 / (1.0 - cos((kx * -2.0))))));
	} else if (t_2 <= 5e-60) {
		tmp = (ky * sin(th)) / sin(kx);
	} else if (t_2 <= 0.0004) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (ky * ky)))) * fma(ky, ((ky * ky) * fma(0.008333333333333333, (ky * ky), -0.16666666666666666)), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(kx, kx, t_1))));
	elseif (t_2 <= 1e-262)
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	elseif (t_2 <= 5e-60)
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	elseif (t_2 <= 0.0004)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(ky * ky)))) * fma(ky, Float64(Float64(ky * ky) * fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666)), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(kx * kx + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-262], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-60], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0004], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $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))))) < -1

   1. Initial program 92.3%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.3

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

   5. Simplified 92.3%

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

   6. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 41.7

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

   8. Simplified 41.7%

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

   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))))) < 1.00000000000000001e-262

   1. Initial program 99.5%

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

   4. Applied egg-rr 85.5%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower--.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.7

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

   7. Simplified 52.7%

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

   if 1.00000000000000001e-262 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-60

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 61.4

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

   5. Simplified 61.4%

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

   if 5.0000000000000001e-60 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000019e-4

   1. Initial program 99.1%

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

   4. Applied egg-rr 77.9%

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

   5. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.4

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

   7. Simplified 79.4%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 79.4

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

   10. Simplified 79.4%

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

   if 4.00000000000000019e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.4

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

   5. Simplified 67.4%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(kx, kx, {\sin ky}^{2}\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-262}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(kx, kx, t\_1\right)}}\\ \mathbf{elif}\;t\_2 \leq 10^{-262}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -1.0)
     (* (* (sin ky) th) (sqrt (/ 1.0 (fma kx kx t_1))))
     (if (<= t_2 1e-262)
       (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5))))
       (if (<= t_2 5e-60)
         (/ (* ky (sin th)) (sin kx))
         (if (<= t_2 0.0004)
           (*
            (sin th)
            (*
             (sqrt (/ 1.0 (fma (- 1.0 (cos (+ kx kx))) 0.5 (* ky ky))))
             (fma
              ky
              (*
               (* ky ky)
               (fma 0.008333333333333333 (* ky ky) -0.16666666666666666))
              ky)))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(kx, kx, t_1)));
	} else if (t_2 <= 1e-262) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else if (t_2 <= 5e-60) {
		tmp = (ky * sin(th)) / sin(kx);
	} else if (t_2 <= 0.0004) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (ky * ky)))) * fma(ky, ((ky * ky) * fma(0.008333333333333333, (ky * ky), -0.16666666666666666)), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(kx, kx, t_1))));
	elseif (t_2 <= 1e-262)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	elseif (t_2 <= 5e-60)
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	elseif (t_2 <= 0.0004)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(ky * ky)))) * fma(ky, Float64(Float64(ky * ky) * fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666)), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(kx * kx + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-262], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-60], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0004], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $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))))) < -1

   1. Initial program 92.3%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.3

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

   5. Simplified 92.3%

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

   6. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 41.7

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

   8. Simplified 41.7%

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

   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))))) < 1.00000000000000001e-262

   1. Initial program 99.5%

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

   4. Applied egg-rr 85.5%

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

   6. Applied egg-rr 10.5%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-fma.f64 10.5

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. lift-+.f64 85.5

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

   8. Applied egg-rr 85.5%

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

   9. Taylor expanded in ky around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

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

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

       4. lower-fma.f64 N/A

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

       5. cos-neg N/A

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

       6. lower-cos.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 52.7

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

   11. Simplified 52.7%

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

   if 1.00000000000000001e-262 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-60

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 61.4

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

   5. Simplified 61.4%

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

   if 5.0000000000000001e-60 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000019e-4

   1. Initial program 99.1%

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

   4. Applied egg-rr 77.9%

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

   5. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.4

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

   7. Simplified 79.4%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 79.4

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

   10. Simplified 79.4%

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

   if 4.00000000000000019e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.4

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

   5. Simplified 67.4%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(kx, kx, {\sin ky}^{2}\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-262}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{\sin ky \cdot th}}\\ \mathbf{elif}\;t\_1 \leq 10^{-262}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -1.0)
     (/
      1.0
      (/ (sqrt (fma -0.5 (cos (* ky -2.0)) (fma kx kx 0.5))) (* (sin ky) th)))
     (if (<= t_1 1e-262)
       (* (sin th) (/ (sin ky) (sqrt (fma -0.5 (cos (* kx -2.0)) 0.5))))
       (if (<= t_1 5e-60)
         (/ (* ky (sin th)) (sin kx))
         (if (<= t_1 0.0004)
           (*
            (sin th)
            (*
             (sqrt (/ 1.0 (fma (- 1.0 (cos (+ kx kx))) 0.5 (* ky ky))))
             (fma
              ky
              (*
               (* ky ky)
               (fma 0.008333333333333333 (* ky ky) -0.16666666666666666))
              ky)))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = 1.0 / (sqrt(fma(-0.5, cos((ky * -2.0)), fma(kx, kx, 0.5))) / (sin(ky) * th));
	} else if (t_1 <= 1e-262) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else if (t_1 <= 5e-60) {
		tmp = (ky * sin(th)) / sin(kx);
	} else if (t_1 <= 0.0004) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (ky * ky)))) * fma(ky, ((ky * ky) * fma(0.008333333333333333, (ky * ky), -0.16666666666666666)), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(1.0 / Float64(sqrt(fma(-0.5, cos(Float64(ky * -2.0)), fma(kx, kx, 0.5))) / Float64(sin(ky) * th)));
	elseif (t_1 <= 1e-262)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	elseif (t_1 <= 5e-60)
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	elseif (t_1 <= 0.0004)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(ky * ky)))) * fma(ky, Float64(Float64(ky * ky) * fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666)), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(1.0 / N[(N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(kx * kx + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-262], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-60], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0004], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $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))))) < -1

   1. Initial program 92.3%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.3

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

   5. Simplified 92.3%

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

   7. Applied egg-rr 70.4%

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

   8. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sin.f64 26.9

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

   10. Simplified 26.9%

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

   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))))) < 1.00000000000000001e-262

   1. Initial program 99.5%

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

   4. Applied egg-rr 85.5%

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

   6. Applied egg-rr 10.5%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-fma.f64 10.5

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. lift-+.f64 85.5

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

   8. Applied egg-rr 85.5%

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

   9. Taylor expanded in ky around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

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

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

       4. lower-fma.f64 N/A

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

       5. cos-neg N/A

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

       6. lower-cos.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 52.7

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

   11. Simplified 52.7%

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

   if 1.00000000000000001e-262 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-60

   1. Initial program 99.4%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 61.4

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

   5. Simplified 61.4%

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

   if 5.0000000000000001e-60 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000019e-4

   1. Initial program 99.1%

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

   4. Applied egg-rr 77.9%

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

   5. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.4

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

   7. Simplified 79.4%

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

   8. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 79.4

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

   10. Simplified 79.4%

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

   if 4.00000000000000019e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.4

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

   5. Simplified 67.4%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{\sin ky \cdot th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-262}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.42:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{\sin ky \cdot th}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{t\_1}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.0004:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.42)
     (/
      1.0
      (/ (sqrt (fma -0.5 (cos (* ky -2.0)) (fma kx kx 0.5))) (* (sin ky) th)))
     (if (<= t_2 5e-60)
       (/ t_1 (sin kx))
       (if (<= t_2 0.0004)
         (* t_1 (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5))))
         (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.42) {
		tmp = 1.0 / (sqrt(fma(-0.5, cos((ky * -2.0)), fma(kx, kx, 0.5))) / (sin(ky) * th));
	} else if (t_2 <= 5e-60) {
		tmp = t_1 / sin(kx);
	} else if (t_2 <= 0.0004) {
		tmp = t_1 * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.42)
		tmp = Float64(1.0 / Float64(sqrt(fma(-0.5, cos(Float64(ky * -2.0)), fma(kx, kx, 0.5))) / Float64(sin(ky) * th)));
	elseif (t_2 <= 5e-60)
		tmp = Float64(t_1 / sin(kx));
	elseif (t_2 <= 0.0004)
		tmp = Float64(t_1 * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.42], N[(1.0 / N[(N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(kx * kx + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-60], N[(t$95$1 / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0004], N[(t$95$1 * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.419999999999999984

   1. Initial program 95.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 55.0

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

   5. Simplified 55.0%

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

   7. Applied egg-rr 43.0%

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

   8. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sin.f64 17.0

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

   10. Simplified 17.0%

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

   if -0.419999999999999984 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-60

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 59.4

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

   5. Simplified 59.4%

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

   if 5.0000000000000001e-60 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000019e-4

   1. Initial program 99.1%

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

   4. Applied egg-rr 77.9%

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

   6. Applied egg-rr 2.6%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-fma.f64 2.6

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. lift-+.f64 77.4

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

   8. Applied egg-rr 77.4%

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

   9. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. metadata-eval N/A

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

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

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

       9. lower-fma.f64 N/A

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

       10. cos-neg N/A

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

       11. lower-cos.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 78.2

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

   11. Simplified 78.2%

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

   if 4.00000000000000019e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.4

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

   5. Simplified 67.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.42:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{\sin ky \cdot th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0004:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.42:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{t\_1}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.0004:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.42)
     (*
      (* (sin ky) th)
      (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) (fma kx kx 0.5)))))
     (if (<= t_2 5e-60)
       (/ t_1 (sin kx))
       (if (<= t_2 0.0004)
         (* t_1 (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5))))
         (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.42) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), fma(kx, kx, 0.5))));
	} else if (t_2 <= 5e-60) {
		tmp = t_1 / sin(kx);
	} else if (t_2 <= 0.0004) {
		tmp = t_1 * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.42)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), fma(kx, kx, 0.5)))));
	elseif (t_2 <= 5e-60)
		tmp = Float64(t_1 / sin(kx));
	elseif (t_2 <= 0.0004)
		tmp = Float64(t_1 * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.42], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(kx * kx + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-60], N[(t$95$1 / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0004], N[(t$95$1 * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.419999999999999984

   1. Initial program 95.5%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 55.0

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

   5. Simplified 55.0%

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

   7. Applied egg-rr 43.0%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

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

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

      11. cos-neg N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-fma.f64 17.0

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

   10. Simplified 17.0%

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

   if -0.419999999999999984 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-60

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 59.4

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

   5. Simplified 59.4%

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

   if 5.0000000000000001e-60 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000019e-4

   1. Initial program 99.1%

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

   4. Applied egg-rr 77.9%

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

   6. Applied egg-rr 2.6%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-fma.f64 2.6

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. lift-+.f64 77.4

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

   8. Applied egg-rr 77.4%

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

   9. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. metadata-eval N/A

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

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

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

       9. lower-fma.f64 N/A

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

       10. cos-neg N/A

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

       11. lower-cos.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 78.2

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

   11. Simplified 78.2%

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

   if 4.00000000000000019e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.4

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

   5. Simplified 67.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.42:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0004:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\frac{t\_1}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.0004:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 5e-60)
     (/ t_1 (sin kx))
     (if (<= t_2 0.0004)
       (* t_1 (sqrt (/ 1.0 (fma -0.5 (cos (* kx -2.0)) 0.5))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 5e-60) {
		tmp = t_1 / sin(kx);
	} else if (t_2 <= 0.0004) {
		tmp = t_1 * sqrt((1.0 / fma(-0.5, cos((kx * -2.0)), 0.5)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 5e-60)
		tmp = Float64(t_1 / sin(kx));
	elseif (t_2 <= 0.0004)
		tmp = Float64(t_1 * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(kx * -2.0)), 0.5))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-60], N[(t$95$1 / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0004], N[(t$95$1 * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-60

   1. Initial program 97.7%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 35.5

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

   5. Simplified 35.5%

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

   if 5.0000000000000001e-60 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000019e-4

   1. Initial program 99.1%

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

   4. Applied egg-rr 77.9%

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

   6. Applied egg-rr 2.6%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-fma.f64 2.6

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. lift-+.f64 77.4

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

   8. Applied egg-rr 77.4%

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

   9. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. metadata-eval N/A

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

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

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

       9. lower-fma.f64 N/A

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

       10. cos-neg N/A

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

       11. lower-cos.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 78.2

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

   11. Simplified 78.2%

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

   if 4.00000000000000019e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.4

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

   5. Simplified 67.4%

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

Alternative 10: 38.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := ky \cdot \sqrt{2}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, 0.08888888888888889, -0.6666666666666666\right), 2\right)}} \cdot t\_2\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t\_2 \cdot \left(th \cdot \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (* ky (sqrt 2.0))))
   (if (<= t_1 5e-60)
     (*
      (sin th)
      (*
       (sqrt
        (/
         1.0
         (*
          (* kx kx)
          (fma
           (* kx kx)
           (fma (* kx kx) 0.08888888888888889 -0.6666666666666666)
           2.0))))
       t_2))
     (if (<= t_1 2e-10)
       (* t_2 (* th (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0)))))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = ky * sqrt(2.0);
	double tmp;
	if (t_1 <= 5e-60) {
		tmp = sin(th) * (sqrt((1.0 / ((kx * kx) * fma((kx * kx), fma((kx * kx), 0.08888888888888889, -0.6666666666666666), 2.0)))) * t_2);
	} else if (t_1 <= 2e-10) {
		tmp = t_2 * (th * sqrt((1.0 / (1.0 - cos((kx * -2.0))))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(ky * sqrt(2.0))
	tmp = 0.0
	if (t_1 <= 5e-60)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(Float64(kx * kx) * fma(Float64(kx * kx), fma(Float64(kx * kx), 0.08888888888888889, -0.6666666666666666), 2.0)))) * t_2));
	elseif (t_1 <= 2e-10)
		tmp = Float64(t_2 * Float64(th * sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-60], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * 0.08888888888888889 + -0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-10], N[(t$95$2 * N[(th * N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-60

   1. Initial program 97.7%

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

   4. Applied egg-rr 76.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 38.9

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

   7. Simplified 38.9%

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

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{{kx}^{2} \cdot \left(2 + {kx}^{2} \cdot \left(\frac{4}{45} \cdot {kx}^{2} - \frac{2}{3}\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{{kx}^{2} \cdot \left(2 + {kx}^{2} \cdot \left(\frac{4}{45} \cdot {kx}^{2} - \frac{2}{3}\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(2 + {kx}^{2} \cdot \left(\frac{4}{45} \cdot {kx}^{2} - \frac{2}{3}\right)\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\left(kx \cdot kx\right)} \cdot \left(2 + {kx}^{2} \cdot \left(\frac{4}{45} \cdot {kx}^{2} - \frac{2}{3}\right)\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \color{blue}{\left({kx}^{2} \cdot \left(\frac{4}{45} \cdot {kx}^{2} - \frac{2}{3}\right) + 2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{4}{45} \cdot {kx}^{2} - \frac{2}{3}, 2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{4}{45} \cdot {kx}^{2} - \frac{2}{3}, 2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{4}{45} \cdot {kx}^{2} - \frac{2}{3}, 2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      8. sub-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{4}{45} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{4}{45}} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right), 2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, {kx}^{2} \cdot \frac{4}{45} + \color{blue}{\frac{-2}{3}}, 2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{4}{45}, \frac{-2}{3}\right)}, 2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{4}{45}, \frac{-2}{3}\right), 2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      13. lower-*.f64 28.9

          \[\leadsto \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, 0.08888888888888889, -0.6666666666666666\right), 2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

   10. Simplified 28.9%

       \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, 0.08888888888888889, -0.6666666666666666\right), 2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

   if 5.0000000000000001e-60 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.1%

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

   4. Applied egg-rr 77.9%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 77.6

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

   7. Simplified 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   9. Applied egg-rr 78.1%

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

   10. Taylor expanded in th around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. cos-neg N/A

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

       5. lower--.f64 N/A

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

       6. cos-neg N/A

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

       7. lower-cos.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 46.1

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

   12. Simplified 46.1%

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

   if 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.4

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

   5. Simplified 67.4%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, 0.08888888888888889, -0.6666666666666666\right), 2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\left(ky \cdot \sqrt{2}\right) \cdot \left(th \cdot \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \sqrt{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\sin th \cdot \left(t\_1 \cdot \sqrt{\frac{0.5}{kx \cdot kx}}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t\_1 \cdot \left(th \cdot \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (sqrt 2.0)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 5e-60)
     (* (sin th) (* t_1 (sqrt (/ 0.5 (* kx kx)))))
     (if (<= t_2 2e-10)
       (* t_1 (* th (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0)))))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * sqrt(2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 5e-60) {
		tmp = sin(th) * (t_1 * sqrt((0.5 / (kx * kx))));
	} else if (t_2 <= 2e-10) {
		tmp = t_1 * (th * sqrt((1.0 / (1.0 - cos((kx * -2.0))))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ky * sqrt(2.0d0)
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_2 <= 5d-60) then
        tmp = sin(th) * (t_1 * sqrt((0.5d0 / (kx * kx))))
    else if (t_2 <= 2d-10) then
        tmp = t_1 * (th * sqrt((1.0d0 / (1.0d0 - cos((kx * (-2.0d0)))))))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = ky * Math.sqrt(2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 5e-60) {
		tmp = Math.sin(th) * (t_1 * Math.sqrt((0.5 / (kx * kx))));
	} else if (t_2 <= 2e-10) {
		tmp = t_1 * (th * Math.sqrt((1.0 / (1.0 - Math.cos((kx * -2.0))))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = ky * math.sqrt(2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= 5e-60:
		tmp = math.sin(th) * (t_1 * math.sqrt((0.5 / (kx * kx))))
	elif t_2 <= 2e-10:
		tmp = t_1 * (th * math.sqrt((1.0 / (1.0 - math.cos((kx * -2.0))))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * sqrt(2.0))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 5e-60)
		tmp = Float64(sin(th) * Float64(t_1 * sqrt(Float64(0.5 / Float64(kx * kx)))));
	elseif (t_2 <= 2e-10)
		tmp = Float64(t_1 * Float64(th * sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = ky * sqrt(2.0);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= 5e-60)
		tmp = sin(th) * (t_1 * sqrt((0.5 / (kx * kx))));
	elseif (t_2 <= 2e-10)
		tmp = t_1 * (th * sqrt((1.0 / (1.0 - cos((kx * -2.0))))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-60], N[(N[Sin[th], $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(0.5 / N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-10], N[(t$95$1 * N[(th * N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-60

   1. Initial program 97.7%

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

   4. Applied egg-rr 76.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 38.9

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

   7. Simplified 38.9%

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

   8. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 28.8

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

   10. Simplified 28.8%

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

   if 5.0000000000000001e-60 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.1%

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

   4. Applied egg-rr 77.9%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 77.6

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

   7. Simplified 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   9. Applied egg-rr 78.1%

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

   10. Taylor expanded in th around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. cos-neg N/A

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

       5. lower--.f64 N/A

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

       6. cos-neg N/A

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

       7. lower-cos.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 46.1

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

   12. Simplified 46.1%

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

   if 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.4

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

   5. Simplified 67.4%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\sin th \cdot \left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{0.5}{kx \cdot kx}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\left(ky \cdot \sqrt{2}\right) \cdot \left(th \cdot \sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\sin th \cdot \left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{0.5}{kx \cdot kx}}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \left(th \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 5e-60)
     (* (sin th) (* (* ky (sqrt 2.0)) (sqrt (/ 0.5 (* kx kx)))))
     (if (<= t_1 2e-10)
       (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (* th (sqrt 2.0))))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-60) {
		tmp = sin(th) * ((ky * sqrt(2.0)) * sqrt((0.5 / (kx * kx))));
	} else if (t_1 <= 2e-10) {
		tmp = sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * (th * sqrt(2.0)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= 5d-60) then
        tmp = sin(th) * ((ky * sqrt(2.0d0)) * sqrt((0.5d0 / (kx * kx))))
    else if (t_1 <= 2d-10) then
        tmp = sqrt((1.0d0 / (1.0d0 - cos((kx * (-2.0d0)))))) * (ky * (th * sqrt(2.0d0)))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-60) {
		tmp = Math.sin(th) * ((ky * Math.sqrt(2.0)) * Math.sqrt((0.5 / (kx * kx))));
	} else if (t_1 <= 2e-10) {
		tmp = Math.sqrt((1.0 / (1.0 - Math.cos((kx * -2.0))))) * (ky * (th * Math.sqrt(2.0)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 5e-60:
		tmp = math.sin(th) * ((ky * math.sqrt(2.0)) * math.sqrt((0.5 / (kx * kx))))
	elif t_1 <= 2e-10:
		tmp = math.sqrt((1.0 / (1.0 - math.cos((kx * -2.0))))) * (ky * (th * math.sqrt(2.0)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 5e-60)
		tmp = Float64(sin(th) * Float64(Float64(ky * sqrt(2.0)) * sqrt(Float64(0.5 / Float64(kx * kx)))));
	elseif (t_1 <= 2e-10)
		tmp = Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * Float64(th * sqrt(2.0))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 5e-60)
		tmp = sin(th) * ((ky * sqrt(2.0)) * sqrt((0.5 / (kx * kx))));
	elseif (t_1 <= 2e-10)
		tmp = sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * (th * sqrt(2.0)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-60], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-10], N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(th * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-60

   1. Initial program 97.7%

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

   4. Applied egg-rr 76.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 38.9

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

   7. Simplified 38.9%

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

   8. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 28.8

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

   10. Simplified 28.8%

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

   if 5.0000000000000001e-60 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.1%

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

   4. Applied egg-rr 77.9%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower--.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 77.6

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

   7. Simplified 77.6%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. cos-neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower--.f64 N/A

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

      11. cos-neg N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   10. Simplified 45.9%

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

   if 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.4

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

   5. Simplified 67.4%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-60}:\\ \;\;\;\;\sin th \cdot \left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{0.5}{kx \cdot kx}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \left(th \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(kx, kx, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0)))
   (if (<= t_1 0.0)
     (sin th)
     (if (<= t_1 1e-9)
       (/
        (sin th)
        (/ (sqrt (fma kx kx (fma (cos (+ ky ky)) -0.5 0.5))) (sin ky)))
       (* (sin th) (* (sin ky) (sqrt (/ 2.0 (- 1.0 (cos (* kx -2.0)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sin(th);
	} else if (t_1 <= 1e-9) {
		tmp = sin(th) / (sqrt(fma(kx, kx, fma(cos((ky + ky)), -0.5, 0.5))) / sin(ky));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((2.0 / (1.0 - cos((kx * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sin(th);
	elseif (t_1 <= 1e-9)
		tmp = Float64(sin(th) / Float64(sqrt(fma(kx, kx, fma(cos(Float64(ky + ky)), -0.5, 0.5))) / sin(ky)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sin[th], $MachinePrecision], If[LessEqual[t$95$1, 1e-9], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(kx * kx + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.0

   1. Initial program 78.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.7

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

   5. Simplified 56.7%

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

   if 0.0 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 1.00000000000000006e-9

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.8

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

   5. Simplified 99.8%

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

   7. Applied egg-rr 91.6%

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

   if 1.00000000000000006e-9 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower--.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.5

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

   7. Simplified 59.5%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;{\sin kx}^{2} \leq 10^{-9}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(kx, kx, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(kx, kx, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0)))
   (if (<= t_1 0.0)
     (sin th)
     (if (<= t_1 1e-9)
       (*
        (sin ky)
        (/ (sin th) (sqrt (fma kx kx (fma (cos (+ ky ky)) -0.5 0.5)))))
       (* (sin th) (* (sin ky) (sqrt (/ 2.0 (- 1.0 (cos (* kx -2.0)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sin(th);
	} else if (t_1 <= 1e-9) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(kx, kx, fma(cos((ky + ky)), -0.5, 0.5))));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((2.0 / (1.0 - cos((kx * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sin(th);
	elseif (t_1 <= 1e-9)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(kx, kx, fma(cos(Float64(ky + ky)), -0.5, 0.5)))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sin[th], $MachinePrecision], If[LessEqual[t$95$1, 1e-9], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(kx * kx + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.0

   1. Initial program 78.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 56.7

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

   5. Simplified 56.7%

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

   if 0.0 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 1.00000000000000006e-9

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.8

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

   5. Simplified 99.8%

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

   7. Applied egg-rr 91.5%

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

   if 1.00000000000000006e-9 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower--.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.5

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 59.5%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;{\sin kx}^{2} \leq 10^{-9}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(kx, kx, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.001)
   (* ky (/ (sin th) (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.001) {
		tmp = ky * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.001d0) then
        tmp = ky * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.001) {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.001:
		tmp = ky * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.001)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.001)
		tmp = ky * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.001], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1e-3

   1. Initial program 97.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right)\right)} + \frac{\sin th}{\sin kx}\right) \]

      3. *-commutative N/A

         \[\leadsto ky \cdot \left(\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right)\right) + \frac{\sin th}{\sin kx}\right) \]

      4. associate-+r+ N/A

         \[\leadsto ky \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2} + \left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto ky \cdot \color{blue}{\left(\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right) + \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2}\right)} \]

   5. Simplified 33.2%

      \[\leadsto \color{blue}{ky \cdot \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right), \frac{\sin th}{\sin kx}, \frac{\left(ky \cdot ky\right) \cdot \left(-0.5 \cdot \sin th\right)}{{\sin kx}^{3}}\right)} \]

   6. Taylor expanded in ky around 0

      \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

      2. lower-sin.f64 N/A

         \[\leadsto ky \cdot \frac{\color{blue}{\sin th}}{\sin kx} \]

      3. lower-sin.f64 35.5

         \[\leadsto ky \cdot \frac{\sin th}{\color{blue}{\sin kx}} \]

   8. Simplified 35.5%

      \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   if 1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 67.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 38.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{0.5}{kx \cdot kx}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.001)
   (* (sin th) (* (* ky (sqrt 2.0)) (sqrt (/ 0.5 (* kx kx)))))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.001) {
		tmp = sin(th) * ((ky * sqrt(2.0)) * sqrt((0.5 / (kx * kx))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.001d0) then
        tmp = sin(th) * ((ky * sqrt(2.0d0)) * sqrt((0.5d0 / (kx * kx))))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.001) {
		tmp = Math.sin(th) * ((ky * Math.sqrt(2.0)) * Math.sqrt((0.5 / (kx * kx))));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.001:
		tmp = math.sin(th) * ((ky * math.sqrt(2.0)) * math.sqrt((0.5 / (kx * kx))))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.001)
		tmp = Float64(sin(th) * Float64(Float64(ky * sqrt(2.0)) * sqrt(Float64(0.5 / Float64(kx * kx)))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.001)
		tmp = sin(th) * ((ky * sqrt(2.0)) * sqrt((0.5 / (kx * kx))));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.001], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1e-3

   1. Initial program 97.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      8. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      10. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \color{blue}{\left(ky \cdot \sqrt{2}\right)}\right) \cdot \sin th \]

      13. lower-sqrt.f64 41.0

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Simplified 41.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\color{blue}{\frac{\frac{1}{2}}{{kx}^{2}}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\frac{1}{2}}{{kx}^{2}}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{\frac{1}{2}}{\color{blue}{kx \cdot kx}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

      3. lower-*.f64 28.6

         \[\leadsto \left(\sqrt{\frac{0.5}{\color{blue}{kx \cdot kx}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

   10. Simplified 28.6%

       \[\leadsto \left(\sqrt{\color{blue}{\frac{0.5}{kx \cdot kx}}} \cdot \left(ky \cdot \sqrt{2}\right)\right) \cdot \sin th \]

   if 1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 67.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \left(\left(ky \cdot \sqrt{2}\right) \cdot \sqrt{\frac{0.5}{kx \cdot kx}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-24}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-24)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (/
    (sin th)
    (/
     (sqrt (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* -0.5 (cos (+ ky ky))))))
     (sin ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 1e-24) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) / (sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (-0.5 * cos((ky + ky)))))) / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 1e-24)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))) / sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 1e-24], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 9.99999999999999924e-25

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 92.7

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 92.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky + kx \cdot kx}}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \sin th \]

      6. lower-hypot.f64 99.8

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   7. Applied egg-rr 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   if 9.99999999999999924e-25 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-24}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-24}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \left(kx + kx\right), -0.5, 0.5\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-24)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (*
    (sin th)
    (/
     (sin ky)
     (sqrt
      (fma (- 1.0 (cos (+ ky ky))) 0.5 (fma (cos (+ kx kx)) -0.5 0.5)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 1e-24) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, fma(cos((kx + kx)), -0.5, 0.5))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 1e-24)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, fma(cos(Float64(kx + kx)), -0.5, 0.5)))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 1e-24], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 9.99999999999999924e-25

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 92.7

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 92.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky + kx \cdot kx}}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \sin th \]

      6. lower-hypot.f64 99.8

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   7. Applied egg-rr 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   if 9.99999999999999924e-25 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   6. Applied egg-rr 12.3%

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}} \cdot \sin th} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \cos \color{blue}{\left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin th \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin th \]

      3. lift-fma.f64 12.3

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}\right)}} \cdot \sin th \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(ky + ky\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      5. flip-+ N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(\frac{ky \cdot ky - ky \cdot ky}{ky - ky}\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{ky \cdot ky} - ky \cdot ky}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{ky \cdot ky - \color{blue}{ky \cdot ky}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      8. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{0}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      9. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{kx \cdot kx - kx \cdot kx}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      10. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      11. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{kx - kx}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      12. flip-+ N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      13. lift-+.f64 99.1

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, -0.5, 0.5\right)\right)}} \cdot \sin th \]

   8. Applied egg-rr 99.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(kx + kx\right), -0.5, 0.5\right)}\right)}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-24}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \left(kx + kx\right), -0.5, 0.5\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 15.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-305}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      4e-305)
   (* -0.16666666666666666 (* th (* th th)))
   th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(th) * (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 4e-305) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = 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(th) * (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))))) <= 4d-305) then
        tmp = (-0.16666666666666666d0) * (th * (th * th))
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(th) * (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))))) <= 4e-305) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(th) * (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))))) <= 4e-305:
		tmp = -0.16666666666666666 * (th * (th * th))
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(th) * Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 4e-305)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(th) * (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 4e-305)
		tmp = -0.16666666666666666 * (th * (th * th));
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-305], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], th]

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)) < 3.99999999999999999e-305

   1. Initial program 96.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 23.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 23.4%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

      5. lower-*.f64 13.8

         \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

   8. Simplified 13.8%

      \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot \left(th \cdot th\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{{th}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot {th}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       6. lower-*.f64 18.7

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 18.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 3.99999999999999999e-305 < (*.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 95.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 28.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 28.9%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

      5. lower-*.f64 15.7

         \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

   8. Simplified 15.7%

      \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

   9. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 16.7%

       \[\leadsto th \cdot \color{blue}{1} \]
   12. Step-by-step derivation

       1. *-rgt-identity 16.7

          \[\leadsto \color{blue}{th} \]

   13. Applied egg-rr 16.7%

       \[\leadsto \color{blue}{th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-305}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\frac{ky \cdot \sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.001)
   (/ (* ky (sin th)) kx)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.001) {
		tmp = (ky * sin(th)) / kx;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.001d0) then
        tmp = (ky * sin(th)) / kx
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.001) {
		tmp = (ky * Math.sin(th)) / kx;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.001:
		tmp = (ky * math.sin(th)) / kx
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.001)
		tmp = Float64(Float64(ky * sin(th)) / kx);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.001)
		tmp = (ky * sin(th)) / kx;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.001], N[(N[(ky * N[Sin[th], $MachinePrecision]), $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))))) < 1e-3

   1. Initial program 97.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 50.3

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 50.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{kx} \]

      3. lower-sin.f64 24.0

         \[\leadsto \frac{ky \cdot \color{blue}{\sin th}}{kx} \]

   8. Simplified 24.0%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]

   if 1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 67.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 36.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.001)
   (* (sin th) (/ ky kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.001) {
		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(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.001d0) 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(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.001) {
		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(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.001:
		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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.001)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.001)
		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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.001], N[(N[Sin[th], $MachinePrecision] * N[(ky / 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))))) < 1e-3

   1. Initial program 97.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 50.3

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 50.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]
   7. Step-by-step derivation

      1. lower-/.f64 24.0

         \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]

   8. Simplified 24.0%

      \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]

   if 1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 67.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 31.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 6 \cdot 10^{-77}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 6e-77)
   (* -0.16666666666666666 (* th (* th th)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 6e-77) {
		tmp = -0.16666666666666666 * (th * (th * 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(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 6d-77) then
        tmp = (-0.16666666666666666d0) * (th * (th * 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(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 6e-77) {
		tmp = -0.16666666666666666 * (th * (th * 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(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 6e-77:
		tmp = -0.16666666666666666 * (th * (th * th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 6e-77)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * 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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 6e-77)
		tmp = -0.16666666666666666 * (th * (th * 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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 6e-77], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 6.00000000000000033e-77

   1. Initial program 97.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 3.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

      5. lower-*.f64 3.4

         \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

   8. Simplified 3.4%

      \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot \left(th \cdot th\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{{th}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot {th}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       6. lower-*.f64 17.5

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 17.5%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 6.00000000000000033e-77 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 59.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-9)
   (* (sin th) (/ (sin ky) (hypot (sin ky) kx)))
   (* (sin th) (* (sin ky) (sqrt (/ 2.0 (- 1.0 (cos (* kx -2.0)))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 1e-9) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((2.0 / (1.0 - cos((kx * -2.0))))));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(kx), 2.0) <= 1e-9) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), kx));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) * Math.sqrt((2.0 / (1.0 - Math.cos((kx * -2.0))))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(kx), 2.0) <= 1e-9:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), kx))
	else:
		tmp = math.sin(th) * (math.sin(ky) * math.sqrt((2.0 / (1.0 - math.cos((kx * -2.0))))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 1e-9)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(kx) ^ 2.0) <= 1e-9)
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), kx));
	else
		tmp = sin(th) * (sin(ky) * sqrt((2.0 / (1.0 - cos((kx * -2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 1e-9], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 1.00000000000000006e-9

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 92.8

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 92.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky + kx \cdot kx}}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \sin th \]

      6. lower-hypot.f64 99.8

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   7. Applied egg-rr 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   if 1.00000000000000006e-9 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{2}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      5. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-*.f64 59.5

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 59.5%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 51.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{\sin ky \cdot th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-152}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (/
    1.0
    (/ (sqrt (fma -0.5 (cos (* ky -2.0)) (fma kx kx 0.5))) (* (sin ky) th)))
   (if (<= (sin ky) 5e-152)
     (* ky (/ (sin th) (sin kx)))
     (if (<= (sin ky) 0.0004)
       (*
        (sin th)
        (*
         (sqrt (/ 1.0 (fma (- 1.0 (cos (+ kx kx))) 0.5 (* ky ky))))
         (fma
          ky
          (*
           (* ky ky)
           (fma 0.008333333333333333 (* ky ky) -0.16666666666666666))
          ky)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = 1.0 / (sqrt(fma(-0.5, cos((ky * -2.0)), fma(kx, kx, 0.5))) / (sin(ky) * th));
	} else if (sin(ky) <= 5e-152) {
		tmp = ky * (sin(th) / sin(kx));
	} else if (sin(ky) <= 0.0004) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (ky * ky)))) * fma(ky, ((ky * ky) * fma(0.008333333333333333, (ky * ky), -0.16666666666666666)), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.002)
		tmp = Float64(1.0 / Float64(sqrt(fma(-0.5, cos(Float64(ky * -2.0)), fma(kx, kx, 0.5))) / Float64(sin(ky) * th)));
	elseif (sin(ky) <= 5e-152)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 0.0004)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(ky * ky)))) * fma(ky, Float64(Float64(ky * ky) * fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666)), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[(1.0 / N[(N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(kx * kx + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-152], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0004], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -2e-3

   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 kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 48.7

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 48.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Applied egg-rr 47.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx, kx, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}{\sin ky \cdot \sin th}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{th \cdot \sin ky} \cdot \sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]
   9. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}{th \cdot \sin ky}}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}}{th \cdot \sin ky}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}{th \cdot \sin ky}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}}{th \cdot \sin ky}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right) + \frac{1}{2}}}}{th \cdot \sin ky}} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left({kx}^{2} + \frac{1}{2}\right)}}}{th \cdot \sin ky}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), {kx}^{2} + \frac{1}{2}\right)}}}{th \cdot \sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      10. cos-neg N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{kx \cdot kx} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left(kx, kx, \frac{1}{2}\right)}\right)}}{th \cdot \sin ky}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, \frac{1}{2}\right)\right)}}{\color{blue}{th \cdot \sin ky}}} \]

      17. lower-sin.f64 18.7

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{th \cdot \color{blue}{\sin ky}}} \]

   10. Simplified 18.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{th \cdot \sin ky}}} \]

   if -2e-3 < (sin.f64 ky) < 4.9999999999999997e-152

   1. Initial program 89.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right)\right)} + \frac{\sin th}{\sin kx}\right) \]

      3. *-commutative N/A

         \[\leadsto ky \cdot \left(\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right)\right) + \frac{\sin th}{\sin kx}\right) \]

      4. associate-+r+ N/A

         \[\leadsto ky \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2} + \left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto ky \cdot \color{blue}{\left(\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right) + \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2}\right)} \]

   5. Simplified 50.9%

      \[\leadsto \color{blue}{ky \cdot \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right), \frac{\sin th}{\sin kx}, \frac{\left(ky \cdot ky\right) \cdot \left(-0.5 \cdot \sin th\right)}{{\sin kx}^{3}}\right)} \]

   6. Taylor expanded in ky around 0

      \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

      2. lower-sin.f64 N/A

         \[\leadsto ky \cdot \frac{\color{blue}{\sin th}}{\sin kx} \]

      3. lower-sin.f64 56.7

         \[\leadsto ky \cdot \frac{\sin th}{\color{blue}{\sin kx}} \]

   8. Simplified 56.7%

      \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   if 4.9999999999999997e-152 < (sin.f64 ky) < 4.00000000000000019e-4

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. lower-*.f64 88.5

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 88.5%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\left(ky \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) + ky \cdot 1\right)}\right) \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \left(ky \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) + \color{blue}{ky}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right), ky\right)}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      8. sub-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, ky\right)\right) \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}\right), ky\right)\right) \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      12. lower-*.f64 88.5

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{ky \cdot ky}, -0.16666666666666666\right), ky\right)\right) \cdot \sin th \]

   10. Simplified 88.5%

       \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)}\right) \cdot \sin th \]

   if 4.00000000000000019e-4 < (sin.f64 ky)

   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 kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 65.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 65.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{\sin ky \cdot th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-152}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 51.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{\sin ky \cdot th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-152}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (/
    1.0
    (/ (sqrt (fma -0.5 (cos (* ky -2.0)) (fma kx kx 0.5))) (* (sin ky) th)))
   (if (<= (sin ky) 5e-152)
     (* ky (/ (sin th) (sin kx)))
     (if (<= (sin ky) 0.0004)
       (*
        (sin th)
        (*
         (sqrt (/ 1.0 (fma (- 1.0 (cos (+ kx kx))) 0.5 (* ky ky))))
         (fma ky (* -0.16666666666666666 (* ky ky)) ky)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = 1.0 / (sqrt(fma(-0.5, cos((ky * -2.0)), fma(kx, kx, 0.5))) / (sin(ky) * th));
	} else if (sin(ky) <= 5e-152) {
		tmp = ky * (sin(th) / sin(kx));
	} else if (sin(ky) <= 0.0004) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (ky * ky)))) * fma(ky, (-0.16666666666666666 * (ky * ky)), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.002)
		tmp = Float64(1.0 / Float64(sqrt(fma(-0.5, cos(Float64(ky * -2.0)), fma(kx, kx, 0.5))) / Float64(sin(ky) * th)));
	elseif (sin(ky) <= 5e-152)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif (sin(ky) <= 0.0004)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(ky * ky)))) * fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[(1.0 / N[(N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[(kx * kx + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-152], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0004], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -2e-3

   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 kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 48.7

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 48.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   7. Applied egg-rr 47.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(kx, kx, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}{\sin ky \cdot \sin th}}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{th \cdot \sin ky} \cdot \sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}} \]
   9. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}{th \cdot \sin ky}}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}}{th \cdot \sin ky}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}{th \cdot \sin ky}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\frac{1}{2} + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right)}}}{th \cdot \sin ky}} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + {kx}^{2}\right) + \frac{1}{2}}}}{th \cdot \sin ky}} \]

      6. associate-+l+ N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right) + \left({kx}^{2} + \frac{1}{2}\right)}}}{th \cdot \sin ky}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), {kx}^{2} + \frac{1}{2}\right)}}}{th \cdot \sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      10. cos-neg N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(-2 \cdot ky\right)}, {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(ky \cdot -2\right)}, {kx}^{2} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{kx \cdot kx} + \frac{1}{2}\right)}}{th \cdot \sin ky}} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \color{blue}{\mathsf{fma}\left(kx, kx, \frac{1}{2}\right)}\right)}}{th \cdot \sin ky}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, \frac{1}{2}\right)\right)}}{\color{blue}{th \cdot \sin ky}}} \]

      17. lower-sin.f64 18.7

          \[\leadsto \frac{1}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{th \cdot \color{blue}{\sin ky}}} \]

   10. Simplified 18.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{th \cdot \sin ky}}} \]

   if -2e-3 < (sin.f64 ky) < 4.9999999999999997e-152

   1. Initial program 89.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right)\right)} + \frac{\sin th}{\sin kx}\right) \]

      3. *-commutative N/A

         \[\leadsto ky \cdot \left(\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right)\right) + \frac{\sin th}{\sin kx}\right) \]

      4. associate-+r+ N/A

         \[\leadsto ky \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2} + \left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto ky \cdot \color{blue}{\left(\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right) + \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2}\right)} \]

   5. Simplified 50.9%

      \[\leadsto \color{blue}{ky \cdot \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right), \frac{\sin th}{\sin kx}, \frac{\left(ky \cdot ky\right) \cdot \left(-0.5 \cdot \sin th\right)}{{\sin kx}^{3}}\right)} \]

   6. Taylor expanded in ky around 0

      \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

      2. lower-sin.f64 N/A

         \[\leadsto ky \cdot \frac{\color{blue}{\sin th}}{\sin kx} \]

      3. lower-sin.f64 56.7

         \[\leadsto ky \cdot \frac{\sin th}{\color{blue}{\sin kx}} \]

   8. Simplified 56.7%

      \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   if 4.9999999999999997e-152 < (sin.f64 ky) < 4.00000000000000019e-4

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. lower-*.f64 88.5

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 88.5%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)\right)}\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \left(ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}\right)\right) \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1\right)}\right) \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \left(ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\frac{-1}{6} \cdot {ky}^{2}}, ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \frac{-1}{6} \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 88.5

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, -0.16666666666666666 \cdot \color{blue}{\left(ky \cdot ky\right)}, ky\right)\right) \cdot \sin th \]

   10. Simplified 88.5%

       \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}\right) \cdot \sin th \]

   if 4.00000000000000019e-4 < (sin.f64 ky)

   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 kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 65.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 65.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), \mathsf{fma}\left(kx, kx, 0.5\right)\right)}}{\sin ky \cdot th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-152}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 46.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-152}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 5e-152)
   (* (sin th) (/ (sin ky) (sin kx)))
   (if (<= (sin ky) 0.0004)
     (*
      (sin th)
      (*
       (sqrt (/ 1.0 (fma (- 1.0 (cos (+ kx kx))) 0.5 (* ky ky))))
       (fma
        ky
        (* (* ky ky) (fma 0.008333333333333333 (* ky ky) -0.16666666666666666))
        ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 5e-152) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (sin(ky) <= 0.0004) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (ky * ky)))) * fma(ky, ((ky * ky) * fma(0.008333333333333333, (ky * ky), -0.16666666666666666)), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 5e-152)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (sin(ky) <= 0.0004)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(ky * ky)))) * fma(ky, Float64(Float64(ky * ky) * fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666)), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-152], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0004], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < 4.9999999999999997e-152

   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 ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 36.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 36.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 4.9999999999999997e-152 < (sin.f64 ky) < 4.00000000000000019e-4

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. lower-*.f64 88.5

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 88.5%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\left(ky \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) + ky \cdot 1\right)}\right) \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \left(ky \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) + \color{blue}{ky}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right), ky\right)}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      8. sub-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, ky\right)\right) \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}\right), ky\right)\right) \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      12. lower-*.f64 88.5

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{ky \cdot ky}, -0.16666666666666666\right), ky\right)\right) \cdot \sin th \]

   10. Simplified 88.5%

       \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)}\right) \cdot \sin th \]

   if 4.00000000000000019e-4 < (sin.f64 ky)

   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 kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 65.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 65.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-152}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 0.0004:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 43.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.22 \cdot 10^{-162}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 0.5:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.22e-162)
   (sin th)
   (if (<= kx 0.5)
     (*
      (sin th)
      (/
       (sin ky)
       (sqrt
        (fma
         (- 1.0 (cos (+ ky ky)))
         0.5
         (*
          (* kx kx)
          (fma
           (* kx kx)
           (fma
            (* kx kx)
            (fma (* kx kx) -0.0031746031746031746 0.044444444444444446)
            -0.3333333333333333)
           1.0))))))
     (* (sin th) (* (sin ky) (sqrt (/ 2.0 (- 1.0 (cos (* kx -2.0))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.22e-162) {
		tmp = sin(th);
	} else if (kx <= 0.5) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, ((kx * kx) * fma((kx * kx), fma((kx * kx), fma((kx * kx), -0.0031746031746031746, 0.044444444444444446), -0.3333333333333333), 1.0)))));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((2.0 / (1.0 - cos((kx * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.22e-162)
		tmp = sin(th);
	elseif (kx <= 0.5)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(kx * kx) * fma(Float64(kx * kx), fma(Float64(kx * kx), fma(Float64(kx * kx), -0.0031746031746031746, 0.044444444444444446), -0.3333333333333333), 1.0))))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.22e-162], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 0.5], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 1.22000000000000012e-162

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 32.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 32.9%

      \[\leadsto \color{blue}{\sin th} \]

   if 1.22000000000000012e-162 < kx < 0.5

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   6. Applied egg-rr 12.5%

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}} \cdot \sin th} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \cos \color{blue}{\left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin th \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin th \]

      3. lift-fma.f64 12.5

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}\right)}} \cdot \sin th \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(ky + ky\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      5. flip-+ N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(\frac{ky \cdot ky - ky \cdot ky}{ky - ky}\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{ky \cdot ky} - ky \cdot ky}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{ky \cdot ky - \color{blue}{ky \cdot ky}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      8. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{0}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      9. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{kx \cdot kx - kx \cdot kx}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      10. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      11. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{kx - kx}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      12. flip-+ N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      13. lift-+.f64 59.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, -0.5, 0.5\right)\right)}} \cdot \sin th \]

   8. Applied egg-rr 59.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(kx + kx\right), -0.5, 0.5\right)}\right)}} \cdot \sin th \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)}\right)}} \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)}\right)}} \cdot \sin th \]

       2. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)\right)}} \cdot \sin th \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)\right)}} \cdot \sin th \]

       4. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\left({kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right) + 1\right)}\right)}} \cdot \sin th \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\mathsf{fma}\left({kx}^{2}, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}, 1\right)}\right)}} \cdot \sin th \]

       6. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}, 1\right)\right)}} \cdot \sin th \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}, 1\right)\right)}} \cdot \sin th \]

       8. sub-neg N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right)\right)}} \cdot \sin th \]

       9. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, {kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) + \color{blue}{\frac{-1}{3}}, 1\right)\right)}} \cdot \sin th \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}, \frac{-1}{3}\right)}, 1\right)\right)}} \cdot \sin th \]

       11. unpow2 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \sin th \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \sin th \]

       13. +-commutative N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{-1}{315} \cdot {kx}^{2} + \frac{2}{45}}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \sin th \]

       14. *-commutative N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{{kx}^{2} \cdot \frac{-1}{315}} + \frac{2}{45}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \sin th \]

       15. lower-fma.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{-1}{315}, \frac{2}{45}\right)}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \sin th \]

       16. unpow2 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{-1}{315}, \frac{2}{45}\right), \frac{-1}{3}\right), 1\right)\right)}} \cdot \sin th \]

       17. lower-*.f64 91.7

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\color{blue}{kx \cdot kx}, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)\right)}} \cdot \sin th \]

   11. Simplified 91.7%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)}\right)}} \cdot \sin th \]

   if 0.5 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{2}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      5. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-*.f64 64.9

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 64.9%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.22 \cdot 10^{-162}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 0.5:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(kx \cdot kx, -0.0031746031746031746, 0.044444444444444446\right), -0.3333333333333333\right), 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 43.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.14 \cdot 10^{-163}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 0.47:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.14e-163)
   (sin th)
   (if (<= kx 0.47)
     (*
      (sin th)
      (/
       (sin ky)
       (sqrt
        (fma
         (- 1.0 (cos (+ ky ky)))
         0.5
         (*
          (* kx kx)
          (fma
           (* kx kx)
           (fma 0.044444444444444446 (* kx kx) -0.3333333333333333)
           1.0))))))
     (* (sin th) (* (sin ky) (sqrt (/ 2.0 (- 1.0 (cos (* kx -2.0))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.14e-163) {
		tmp = sin(th);
	} else if (kx <= 0.47) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, ((kx * kx) * fma((kx * kx), fma(0.044444444444444446, (kx * kx), -0.3333333333333333), 1.0)))));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((2.0 / (1.0 - cos((kx * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.14e-163)
		tmp = sin(th);
	elseif (kx <= 0.47)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(kx * kx) * fma(Float64(kx * kx), fma(0.044444444444444446, Float64(kx * kx), -0.3333333333333333), 1.0))))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.14e-163], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 0.47], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(kx * kx), $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(0.044444444444444446 * N[(kx * kx), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 1.14000000000000005e-163

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 32.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 32.9%

      \[\leadsto \color{blue}{\sin th} \]

   if 1.14000000000000005e-163 < kx < 0.46999999999999997

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   6. Applied egg-rr 12.5%

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}} \cdot \sin th} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \cos \color{blue}{\left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin th \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin th \]

      3. lift-fma.f64 12.5

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}\right)}} \cdot \sin th \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(ky + ky\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      5. flip-+ N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(\frac{ky \cdot ky - ky \cdot ky}{ky - ky}\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{ky \cdot ky} - ky \cdot ky}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{ky \cdot ky - \color{blue}{ky \cdot ky}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      8. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{0}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      9. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{kx \cdot kx - kx \cdot kx}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      10. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      11. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{kx - kx}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      12. flip-+ N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      13. lift-+.f64 59.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, -0.5, 0.5\right)\right)}} \cdot \sin th \]

   8. Applied egg-rr 59.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(kx + kx\right), -0.5, 0.5\right)}\right)}} \cdot \sin th \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)}\right)}} \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)}\right)}} \cdot \sin th \]

       2. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}} \cdot \sin th \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}} \cdot \sin th \]

       4. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\left({kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) + 1\right)}\right)}} \cdot \sin th \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\mathsf{fma}\left({kx}^{2}, \frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, 1\right)}\right)}} \cdot \sin th \]

       6. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, 1\right)\right)}} \cdot \sin th \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\color{blue}{kx \cdot kx}, \frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, 1\right)\right)}} \cdot \sin th \]

       8. sub-neg N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{\frac{2}{45} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right)\right)}} \cdot \sin th \]

       9. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \frac{2}{45} \cdot {kx}^{2} + \color{blue}{\frac{-1}{3}}, 1\right)\right)}} \cdot \sin th \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \color{blue}{\mathsf{fma}\left(\frac{2}{45}, {kx}^{2}, \frac{-1}{3}\right)}, 1\right)\right)}} \cdot \sin th \]

       11. unpow2 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(\frac{2}{45}, \color{blue}{kx \cdot kx}, \frac{-1}{3}\right), 1\right)\right)}} \cdot \sin th \]

       12. lower-*.f64 91.5

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(0.044444444444444446, \color{blue}{kx \cdot kx}, -0.3333333333333333\right), 1\right)\right)}} \cdot \sin th \]

   11. Simplified 91.5%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), 1\right)}\right)}} \cdot \sin th \]

   if 0.46999999999999997 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{2}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      5. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-*.f64 64.9

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 64.9%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.14 \cdot 10^{-163}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 0.47:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(kx \cdot kx, \mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 43.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.15 \cdot 10^{-165}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 0.09:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.15e-165)
   (sin th)
   (if (<= kx 0.09)
     (*
      (sin th)
      (/
       (sin ky)
       (sqrt
        (fma
         (- 1.0 (cos (+ ky ky)))
         0.5
         (* (* kx kx) (fma -0.3333333333333333 (* kx kx) 1.0))))))
     (* (sin th) (* (sin ky) (sqrt (/ 2.0 (- 1.0 (cos (* kx -2.0))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.15e-165) {
		tmp = sin(th);
	} else if (kx <= 0.09) {
		tmp = sin(th) * (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, ((kx * kx) * fma(-0.3333333333333333, (kx * kx), 1.0)))));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((2.0 / (1.0 - cos((kx * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.15e-165)
		tmp = sin(th);
	elseif (kx <= 0.09)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(kx * kx) * fma(-0.3333333333333333, Float64(kx * kx), 1.0))))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.15e-165], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 0.09], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(kx * kx), $MachinePrecision] * N[(-0.3333333333333333 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 1.15e-165

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 32.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 32.9%

      \[\leadsto \color{blue}{\sin th} \]

   if 1.15e-165 < kx < 0.089999999999999997

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   6. Applied egg-rr 12.5%

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}} \cdot \sin th} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \cos \color{blue}{\left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin th \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin th \]

      3. lift-fma.f64 12.5

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}\right)}} \cdot \sin th \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(ky + ky\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      5. flip-+ N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(\frac{ky \cdot ky - ky \cdot ky}{ky - ky}\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{ky \cdot ky} - ky \cdot ky}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{ky \cdot ky - \color{blue}{ky \cdot ky}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      8. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{0}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      9. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{kx \cdot kx - kx \cdot kx}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      10. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      11. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{kx - kx}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      12. flip-+ N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      13. lift-+.f64 59.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, -0.5, 0.5\right)\right)}} \cdot \sin th \]

   8. Applied egg-rr 59.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(kx + kx\right), -0.5, 0.5\right)}\right)}} \cdot \sin th \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)}\right)}} \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)}\right)}} \cdot \sin th \]

       2. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right)}} \cdot \sin th \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\left(kx \cdot kx\right)} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)\right)}} \cdot \sin th \]

       4. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {kx}^{2} + 1\right)}\right)}} \cdot \sin th \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {kx}^{2}, 1\right)}\right)}} \cdot \sin th \]

       6. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(\frac{-1}{3}, \color{blue}{kx \cdot kx}, 1\right)\right)}} \cdot \sin th \]

       7. lower-*.f64 91.0

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(-0.3333333333333333, \color{blue}{kx \cdot kx}, 1\right)\right)}} \cdot \sin th \]

   11. Simplified 91.0%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\left(kx \cdot kx\right) \cdot \mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right)}\right)}} \cdot \sin th \]

   if 0.089999999999999997 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      2. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{2}{\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      5. cos-neg N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \color{blue}{\cos \left(-2 \cdot kx\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-*.f64 64.9

         \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \color{blue}{\left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 64.9%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.15 \cdot 10^{-165}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 0.09:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(kx \cdot kx\right) \cdot \mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 44.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.2 \cdot 10^{-152}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;ky \leq 0.0024:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.2e-152)
   (* ky (/ (sin th) (sin kx)))
   (if (<= ky 0.0024)
     (*
      (sin th)
      (*
       (sqrt (/ 1.0 (fma (- 1.0 (cos (+ kx kx))) 0.5 (* ky ky))))
       (fma
        ky
        (* (* ky ky) (fma 0.008333333333333333 (* ky ky) -0.16666666666666666))
        ky)))
     (* (sin th) (/ (sin ky) (sqrt (* 0.5 (- 1.0 (cos (* ky -2.0))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.2e-152) {
		tmp = ky * (sin(th) / sin(kx));
	} else if (ky <= 0.0024) {
		tmp = sin(th) * (sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (ky * ky)))) * fma(ky, ((ky * ky) * fma(0.008333333333333333, (ky * ky), -0.16666666666666666)), ky));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt((0.5 * (1.0 - cos((ky * -2.0))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.2e-152)
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	elseif (ky <= 0.0024)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(ky * ky)))) * fma(ky, Float64(Float64(ky * ky) * fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666)), ky)));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(ky * -2.0)))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.2e-152], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 0.0024], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[(N[(ky * ky), $MachinePrecision] * N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 4.19999999999999998e-152

   1. Initial program 93.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto ky \cdot \left(\color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right)\right)} + \frac{\sin th}{\sin kx}\right) \]

      3. *-commutative N/A

         \[\leadsto ky \cdot \left(\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2}} + {ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right)\right) + \frac{\sin th}{\sin kx}\right) \]

      4. associate-+r+ N/A

         \[\leadsto ky \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2} + \left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto ky \cdot \color{blue}{\left(\left({ky}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right) + \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}}\right) \cdot {ky}^{2}\right)} \]

   5. Simplified 31.4%

      \[\leadsto \color{blue}{ky \cdot \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right), \frac{\sin th}{\sin kx}, \frac{\left(ky \cdot ky\right) \cdot \left(-0.5 \cdot \sin th\right)}{{\sin kx}^{3}}\right)} \]

   6. Taylor expanded in ky around 0

      \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

      2. lower-sin.f64 N/A

         \[\leadsto ky \cdot \frac{\color{blue}{\sin th}}{\sin kx} \]

      3. lower-sin.f64 35.0

         \[\leadsto ky \cdot \frac{\sin th}{\color{blue}{\sin kx}} \]

   8. Simplified 35.0%

      \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   if 4.19999999999999998e-152 < ky < 0.00239999999999999979

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{{ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      2. lower-*.f64 88.5

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   7. Simplified 88.5%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \color{blue}{ky \cdot ky}\right)}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\left(ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \left(ky \cdot \color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\left(ky \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) + ky \cdot 1\right)}\right) \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \left(ky \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) + \color{blue}{ky}\right)\right) \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right), ky\right)}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right), ky\right)\right) \cdot \sin th \]

      8. sub-neg N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, ky\right)\right) \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}\right), ky\right)\right) \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, ky\right)\right) \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), \frac{1}{2}, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), ky\right)\right) \cdot \sin th \]

      12. lower-*.f64 88.5

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{ky \cdot ky}, -0.16666666666666666\right), ky\right)\right) \cdot \sin th \]

   10. Simplified 88.5%

       \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)}\right) \cdot \sin th \]

   if 0.00239999999999999979 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin ky\right)} \cdot \sin th \]

   6. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}} \cdot \sin th} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \cos \color{blue}{\left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin th \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{-1}{2} + \frac{1}{2}\right)}} \cdot \sin th \]

      3. lift-fma.f64 21.4

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}\right)}} \cdot \sin th \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(ky + ky\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      5. flip-+ N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(\frac{ky \cdot ky - ky \cdot ky}{ky - ky}\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{ky \cdot ky} - ky \cdot ky}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{ky \cdot ky - \color{blue}{ky \cdot ky}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      8. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{0}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      9. +-inverses N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{\color{blue}{kx \cdot kx - kx \cdot kx}}{ky - ky}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      10. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{0}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      11. +-inverses N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \left(\frac{kx \cdot kx - kx \cdot kx}{\color{blue}{kx - kx}}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      12. flip-+ N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}} \cdot \sin th \]

      13. lift-+.f64 99.4

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \mathsf{fma}\left(\cos \color{blue}{\left(kx + kx\right)}, -0.5, 0.5\right)\right)}} \cdot \sin th \]

   8. Applied egg-rr 99.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \color{blue}{\mathsf{fma}\left(\cos \left(kx + kx\right), -0.5, 0.5\right)}\right)}} \cdot \sin th \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

       2. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

       3. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right)}} \cdot \sin th \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right)}} \cdot \sin th \]

       5. cos-neg N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \left(1 - \color{blue}{\cos \left(-2 \cdot ky\right)}\right)}} \cdot \sin th \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \left(1 - \color{blue}{\cos \left(-2 \cdot ky\right)}\right)}} \cdot \sin th \]

       7. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \color{blue}{\left(ky \cdot -2\right)}\right)}} \cdot \sin th \]

       8. lower-*.f64 63.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \color{blue}{\left(ky \cdot -2\right)}\right)}} \cdot \sin th \]

   11. Simplified 63.6%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 4.2 \cdot 10^{-152}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;ky \leq 0.0024:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, ky \cdot ky\right)}} \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(ky \cdot -2\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 13.4% accurate, 632.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return 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 = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 96.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 26.0

      \[\leadsto \color{blue}{\sin th} \]

5. Simplified 26.0%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto th \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{2}, 1\right)} \]

   4. unpow2 N/A

      \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot th}, 1\right) \]

   5. lower-*.f64 14.7

      \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, 1\right) \]

8. Simplified 14.7%

   \[\leadsto \color{blue}{th \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)} \]

9. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{1} \]
10. Step-by-step derivation

11. Simplified 15.1%

    \[\leadsto th \cdot \color{blue}{1} \]
12. Step-by-step derivation

    1. *-rgt-identity 15.1

       \[\leadsto \color{blue}{th} \]

13. Applied egg-rr 15.1%

    \[\leadsto \color{blue}{th} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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)))