Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.4% → 99.7%

Time: 13.3s

Alternatives: 29

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% 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 93.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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 rewrites 99.6%

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

Alternative 2: 78.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ t_4 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_4}}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_4\right)}}\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_1}}\\ \mathbf{elif}\;t\_3 \leq 0.99996:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_1 t_2))))
        (t_4 (fma -0.5 (cos (* ky -2.0)) 0.5)))
   (if (<= t_3 -0.98)
     (* (sin ky) (/ (sin th) (sqrt t_4)))
     (if (<= t_3 -0.2)
       (*
        (* (sin ky) th)
        (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_4))))
       (if (<= t_3 0.2)
         (* (sin th) (/ (sin ky) (sqrt t_1)))
         (if (<= t_3 0.99996)
           (*
            (*
             (sin ky)
             (sqrt
              (/
               1.0
               (fma
                (- 1.0 (cos (+ kx kx)))
                0.5
                (+ 0.5 (* (cos (+ ky ky)) -0.5))))))
            (fma th (* -0.16666666666666666 (* th th)) th))
           (if (<= t_3 1.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_2) 1.0))
             (* (sin th) (/ (sin ky) (hypot (sin kx) ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((t_1 + t_2));
	double t_4 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double tmp;
	if (t_3 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(t_4));
	} else if (t_3 <= -0.2) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_4)));
	} else if (t_3 <= 0.2) {
		tmp = sin(th) * (sin(ky) / sqrt(t_1));
	} else if (t_3 <= 0.99996) {
		tmp = (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (cos((ky + ky)) * -0.5)))))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else if (t_3 <= 1.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_2), 1.0);
	} else {
		tmp = sin(th) * (sin(ky) / hypot(sin(kx), ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + t_2)))
	t_4 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_4)));
	elseif (t_3 <= -0.2)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_4))));
	elseif (t_3 <= 0.2)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_1)));
	elseif (t_3 <= 0.99996)
		tmp = Float64(Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(cos(Float64(ky + ky)) * -0.5)))))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	elseif (t_3 <= 1.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_2), 1.0));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(kx), ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.2], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.99996], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 80.1%

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 62.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 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 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in ky around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 94.1

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

   8. Applied rewrites 94.1%

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

   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))))) < 0.99995999999999996

   1. Initial program 99.5%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 41.0

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

   7. Applied rewrites 41.0%

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

   if 0.99995999999999996 < (/.f64 (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 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 10.3%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 10.3

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

   5. Applied rewrites 10.3%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2}}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.99996:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ t_4 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_4}}\\ \mathbf{elif}\;t\_3 \leq -0.04:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_4\right)}}\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.99996:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{t\_2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ (sin ky) (hypot (sin kx) ky))))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2))))
        (t_4 (fma -0.5 (cos (* ky -2.0)) 0.5)))
   (if (<= t_3 -0.98)
     (* (sin ky) (/ (sin th) (sqrt t_4)))
     (if (<= t_3 -0.04)
       (*
        (* (sin ky) th)
        (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_4))))
       (if (<= t_3 0.2)
         t_1
         (if (<= t_3 0.99996)
           (*
            (*
             (sin ky)
             (sqrt
              (/
               1.0
               (fma
                (- 1.0 (cos (+ kx kx)))
                0.5
                (+ 0.5 (* (cos (+ ky ky)) -0.5))))))
            (fma th (* -0.16666666666666666 (* th th)) th))
           (if (<= t_3 1.0)
             (* (sin th) (fma (* kx kx) (/ -0.5 t_2) 1.0))
             t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(sin(kx), ky));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double t_4 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double tmp;
	if (t_3 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(t_4));
	} else if (t_3 <= -0.04) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_4)));
	} else if (t_3 <= 0.2) {
		tmp = t_1;
	} else if (t_3 <= 0.99996) {
		tmp = (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (cos((ky + ky)) * -0.5)))))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else if (t_3 <= 1.0) {
		tmp = sin(th) * fma((kx * kx), (-0.5 / t_2), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(sin(kx), ky)))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	t_4 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_4)));
	elseif (t_3 <= -0.04)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_4))));
	elseif (t_3 <= 0.2)
		tmp = t_1;
	elseif (t_3 <= 0.99996)
		tmp = Float64(Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(cos(Float64(ky + ky)) * -0.5)))))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	elseif (t_3 <= 1.0)
		tmp = Float64(sin(th) * fma(Float64(kx * kx), Float64(-0.5 / t_2), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.04], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.2], t$95$1, If[LessEqual[t$95$3, 0.99996], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1.0], N[(N[Sin[th], $MachinePrecision] * N[(N[(kx * kx), $MachinePrecision] * N[(-0.5 / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 80.1%

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0400000000000000008

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 59.0%

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

   if -0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001 or 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. Initial program 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 90.5

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

   5. Applied rewrites 90.5%

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

      1. lift-sin.f64 N/A

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

      2. pow2 N/A

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

      3. lower-hypot.f64 95.0

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

   7. Applied rewrites 95.0%

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

   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))))) < 0.99995999999999996

   1. Initial program 99.5%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 41.0

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

   7. Applied rewrites 41.0%

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

   if 0.99995999999999996 < (/.f64 (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 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.04:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.99996:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1:\\ \;\;\;\;\sin th \cdot \mathsf{fma}\left(kx \cdot kx, \frac{-0.5}{{\sin ky}^{2}}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ \mathbf{if}\;t\_2 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_3\right)}}\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{t\_1}}\\ \mathbf{elif}\;t\_2 \leq 0.99996:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0)))))
        (t_3 (fma -0.5 (cos (* ky -2.0)) 0.5)))
   (if (<= t_2 -0.98)
     (* (sin ky) (/ (sin th) (sqrt t_3)))
     (if (<= t_2 -0.2)
       (*
        (* (sin ky) th)
        (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_3))))
       (if (<= t_2 0.2)
         (* (sin th) (/ (sin ky) (sqrt t_1)))
         (if (<= t_2 0.99996)
           (*
            (*
             (sin ky)
             (sqrt
              (/
               1.0
               (fma
                (- 1.0 (cos (+ kx kx)))
                0.5
                (+ 0.5 (* (cos (+ ky ky)) -0.5))))))
            (fma th (* -0.16666666666666666 (* th th)) th))
           (*
            (sin th)
            (/
             (sin ky)
             (hypot
              (sin ky)
              (fma kx (* -0.16666666666666666 (* kx kx)) kx))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double t_3 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double tmp;
	if (t_2 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(t_3));
	} else if (t_2 <= -0.2) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_3)));
	} else if (t_2 <= 0.2) {
		tmp = sin(th) * (sin(ky) / sqrt(t_1));
	} else if (t_2 <= 0.99996) {
		tmp = (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (cos((ky + ky)) * -0.5)))))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	t_3 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	tmp = 0.0
	if (t_2 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_3)));
	elseif (t_2 <= -0.2)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_3))));
	elseif (t_2 <= 0.2)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(t_1)));
	elseif (t_2 <= 0.99996)
		tmp = Float64(Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(cos(Float64(ky + ky)) * -0.5)))))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$2, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.99996], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 80.1%

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 62.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 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 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in ky around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 94.1

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

   8. Applied rewrites 94.1%

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

   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))))) < 0.99995999999999996

   1. Initial program 99.5%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 41.0

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

   7. Applied rewrites 41.0%

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

   if 0.99995999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2}}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.99996:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ \mathbf{if}\;t\_2 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_2 \leq -0.04:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_3\right)}}\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{t\_1 + ky \cdot ky}}\\ \mathbf{elif}\;t\_2 \leq 0.95:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0)))))
        (t_3 (fma -0.5 (cos (* ky -2.0)) 0.5)))
   (if (<= t_2 -0.98)
     (* (sin ky) (/ (sin th) (sqrt t_3)))
     (if (<= t_2 -0.04)
       (*
        (* (sin ky) th)
        (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_3))))
       (if (<= t_2 0.01)
         (*
          (sin th)
          (/
           (fma ky (* -0.16666666666666666 (* ky ky)) ky)
           (sqrt (+ t_1 (* ky ky)))))
         (if (<= t_2 0.95)
           (*
            (*
             (sin ky)
             (sqrt
              (/
               1.0
               (fma
                (- 1.0 (cos (+ kx kx)))
                0.5
                (+ 0.5 (* (cos (+ ky ky)) -0.5))))))
            (fma th (* -0.16666666666666666 (* th th)) th))
           (/ 1.0 (/ (sin ky) (* (sin ky) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double t_3 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double tmp;
	if (t_2 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(t_3));
	} else if (t_2 <= -0.04) {
		tmp = (sin(ky) * th) * sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_3)));
	} else if (t_2 <= 0.01) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sqrt((t_1 + (ky * ky))));
	} else if (t_2 <= 0.95) {
		tmp = (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (cos((ky + ky)) * -0.5)))))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else {
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	t_3 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	tmp = 0.0
	if (t_2 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_3)));
	elseif (t_2 <= -0.04)
		tmp = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_3))));
	elseif (t_2 <= 0.01)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sqrt(Float64(t_1 + Float64(ky * ky)))));
	elseif (t_2 <= 0.95)
		tmp = Float64(Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(cos(Float64(ky + ky)) * -0.5)))))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	else
		tmp = Float64(1.0 / Float64(sin(ky) / Float64(sin(ky) * sin(th))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[t$95$2, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.04], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.01], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.95], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 80.1%

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0400000000000000008

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 59.0%

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

   if -0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.94999999999999996

   1. Initial program 99.5%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 38.0

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

   7. Applied rewrites 38.0%

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

   if 0.94999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 5.8

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

   5. Applied rewrites 5.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 5.8

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

   7. Applied rewrites 5.8%

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

   9. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.04:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.95:\\ \;\;\;\;\left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\right) \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ t_4 := \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_3\right)}}\\ \mathbf{if}\;t\_2 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_2 \leq -0.04:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot t\_4\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{t\_1 + ky \cdot ky}}\\ \mathbf{elif}\;t\_2 \leq 0.95:\\ \;\;\;\;\sin ky \cdot \left(th \cdot t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0)))))
        (t_3 (fma -0.5 (cos (* ky -2.0)) 0.5))
        (t_4 (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_3)))))
   (if (<= t_2 -0.98)
     (* (sin ky) (/ (sin th) (sqrt t_3)))
     (if (<= t_2 -0.04)
       (* (* (sin ky) th) t_4)
       (if (<= t_2 0.01)
         (*
          (sin th)
          (/
           (fma ky (* -0.16666666666666666 (* ky ky)) ky)
           (sqrt (+ t_1 (* ky ky)))))
         (if (<= t_2 0.95)
           (* (sin ky) (* th t_4))
           (/ 1.0 (/ (sin ky) (* (sin ky) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double t_3 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double t_4 = sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_3)));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(t_3));
	} else if (t_2 <= -0.04) {
		tmp = (sin(ky) * th) * t_4;
	} else if (t_2 <= 0.01) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sqrt((t_1 + (ky * ky))));
	} else if (t_2 <= 0.95) {
		tmp = sin(ky) * (th * t_4);
	} else {
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	t_3 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	t_4 = sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_3)))
	tmp = 0.0
	if (t_2 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_3)));
	elseif (t_2 <= -0.04)
		tmp = Float64(Float64(sin(ky) * th) * t_4);
	elseif (t_2 <= 0.01)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sqrt(Float64(t_1 + Float64(ky * ky)))));
	elseif (t_2 <= 0.95)
		tmp = Float64(sin(ky) * Float64(th * t_4));
	else
		tmp = Float64(1.0 / Float64(sin(ky) / Float64(sin(ky) * sin(th))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.04], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$2, 0.01], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.95], N[(N[Sin[ky], $MachinePrecision] * N[(th * t$95$4), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 80.1%

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0400000000000000008

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 59.0%

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

   if -0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.94999999999999996

   1. Initial program 99.5%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower--.f64 N/A

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

      10. cos-neg N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

   7. Applied rewrites 38.6%

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

   if 0.94999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 5.8

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

   5. Applied rewrites 5.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 5.8

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

   7. Applied rewrites 5.8%

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

   9. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.04:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.95:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\\ t_4 := \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), t\_3\right)}}\\ \mathbf{if}\;t\_2 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_2 \leq -0.04:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{t\_1 + ky \cdot ky}}\\ \mathbf{elif}\;t\_2 \leq 0.95:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0)))))
        (t_3 (fma -0.5 (cos (* ky -2.0)) 0.5))
        (t_4
         (*
          (* (sin ky) th)
          (sqrt (/ 1.0 (fma 0.5 (- 1.0 (cos (* kx -2.0))) t_3))))))
   (if (<= t_2 -0.98)
     (* (sin ky) (/ (sin th) (sqrt t_3)))
     (if (<= t_2 -0.04)
       t_4
       (if (<= t_2 0.01)
         (*
          (sin th)
          (/
           (fma ky (* -0.16666666666666666 (* ky ky)) ky)
           (sqrt (+ t_1 (* ky ky)))))
         (if (<= t_2 0.95) t_4 (/ 1.0 (/ (sin ky) (* (sin ky) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double t_3 = fma(-0.5, cos((ky * -2.0)), 0.5);
	double t_4 = (sin(ky) * th) * sqrt((1.0 / fma(0.5, (1.0 - cos((kx * -2.0))), t_3)));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(t_3));
	} else if (t_2 <= -0.04) {
		tmp = t_4;
	} else if (t_2 <= 0.01) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sqrt((t_1 + (ky * ky))));
	} else if (t_2 <= 0.95) {
		tmp = t_4;
	} else {
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	t_3 = fma(-0.5, cos(Float64(ky * -2.0)), 0.5)
	t_4 = Float64(Float64(sin(ky) * th) * sqrt(Float64(1.0 / fma(0.5, Float64(1.0 - cos(Float64(kx * -2.0))), t_3))))
	tmp = 0.0
	if (t_2 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_3)));
	elseif (t_2 <= -0.04)
		tmp = t_4;
	elseif (t_2 <= 0.01)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sqrt(Float64(t_1 + Float64(ky * ky)))));
	elseif (t_2 <= 0.95)
		tmp = t_4;
	else
		tmp = Float64(1.0 / Float64(sin(ky) / Float64(sin(ky) * sin(th))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(0.5 * N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.04], t$95$4, If[LessEqual[t$95$2, 0.01], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.95], t$95$4, N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 80.1%

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0400000000000000008 or 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.94999999999999996

   1. Initial program 99.3%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 50.7%

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

   if -0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

   if 0.94999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 5.8

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

   5. Applied rewrites 5.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 5.8

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

   7. Applied rewrites 5.8%

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

   9. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.04:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.95:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(0.5, 1 - \cos \left(kx \cdot -2\right), \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.7:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{-223}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.7)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 1.5e-223)
       (*
        (sin ky)
        (/ (sin th) (* (sqrt (- 1.0 (cos (* kx -2.0)))) (sqrt 0.5))))
       (if (<= t_1 0.05)
         (* (sin ky) (/ (sin th) (sin kx)))
         (/ 1.0 (/ (sin ky) (* (sin ky) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.7) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 1.5e-223) {
		tmp = sin(ky) * (sin(th) / (sqrt((1.0 - cos((kx * -2.0)))) * sqrt(0.5)));
	} else if (t_1 <= 0.05) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.7)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 1.5e-223)
		tmp = Float64(sin(ky) * Float64(sin(th) / Float64(sqrt(Float64(1.0 - cos(Float64(kx * -2.0)))) * sqrt(0.5))));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = Float64(1.0 / Float64(sin(ky) / Float64(sin(ky) * sin(th))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.7], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5e-223], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))) < -0.69999999999999996

   1. Initial program 85.0%

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 44.3

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

   7. Applied rewrites 44.3%

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

   if -0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.49999999999999996e-223

   1. Initial program 99.2%

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. cos-neg N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 54.7

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

   7. Applied rewrites 54.7%

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

   if 1.49999999999999996e-223 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 73.1

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

   7. Applied rewrites 73.1%

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

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

   1. Initial program 93.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 7.2

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

   5. Applied rewrites 7.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 7.2

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

   7. Applied rewrites 7.2%

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

   9. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.7:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1.5 \cdot 10^{-223}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{1 - \cos \left(kx \cdot -2\right)} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.7:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{-223}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.7)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 1.5e-223)
       (* (sin th) (* (sin ky) (sqrt (/ 2.0 (- 1.0 (cos (* kx -2.0)))))))
       (if (<= t_1 0.05)
         (* (sin ky) (/ (sin th) (sin kx)))
         (/ 1.0 (/ (sin ky) (* (sin ky) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.7) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 1.5e-223) {
		tmp = sin(th) * (sin(ky) * sqrt((2.0 / (1.0 - cos((kx * -2.0))))));
	} else if (t_1 <= 0.05) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.7)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 1.5e-223)
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(kx * -2.0)))))));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = Float64(1.0 / Float64(sin(ky) / Float64(sin(ky) * sin(th))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.7], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5e-223], 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$1, 0.05], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))) < -0.69999999999999996

   1. Initial program 85.0%

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 44.3

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

   7. Applied rewrites 44.3%

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

   if -0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.49999999999999996e-223

   1. Initial program 99.2%

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in 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 54.7

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

   7. Applied rewrites 54.7%

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

   if 1.49999999999999996e-223 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 73.1

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

   7. Applied rewrites 73.1%

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

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

   1. Initial program 93.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 7.2

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

   5. Applied rewrites 7.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 7.2

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

   7. Applied rewrites 7.2%

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

   9. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.7:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1.5 \cdot 10^{-223}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{2}{1 - \cos \left(kx \cdot -2\right)}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{ky}{\sin kx}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot \left(th \cdot th\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ ky (sin kx))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.2)
     (*
      (* (sin ky) (fma -0.16666666666666666 (* th (* th th)) th))
      (sqrt (/ 1.0 (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_2 0.01) t_1 (if (<= t_2 2.0) (sin th) t_1)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (ky / sin(kx));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.2) {
		tmp = (sin(ky) * fma(-0.16666666666666666, (th * (th * th)), th)) * sqrt((1.0 / fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_2 <= 0.01) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(ky / sin(kx)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.2)
		tmp = Float64(Float64(sin(ky) * fma(-0.16666666666666666, Float64(th * Float64(th * th)), th)) * sqrt(Float64(1.0 / fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_2 <= 0.01)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] * N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.01], t$95$1, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 88.0%

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

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in th around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 40.3

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

   7. Applied rewrites 40.3%

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

   8. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. lower-fma.f64 N/A

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

      17. cos-neg N/A

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

      18. lower-cos.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 18.7

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

   10. Applied rewrites 18.7%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 95.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 98.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 64.8

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

   5. Applied rewrites 64.8%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot \left(th \cdot th\right), th\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.04:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{t\_1 + ky \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.04)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_2 0.01)
       (*
        (sin th)
        (/
         (fma ky (* -0.16666666666666666 (* ky ky)) ky)
         (sqrt (+ t_1 (* ky ky)))))
       (/ 1.0 (/ (sin ky) (* (sin ky) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.04) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_2 <= 0.01) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sqrt((t_1 + (ky * ky))));
	} else {
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.04)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_2 <= 0.01)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sqrt(Float64(t_1 + Float64(ky * ky)))));
	else
		tmp = Float64(1.0 / Float64(sin(ky) / Float64(sin(ky) * sin(th))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.04], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.01], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0400000000000000008

   1. Initial program 88.2%

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 38.5

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

   7. Applied rewrites 38.5%

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

   if -0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

   if 0.0100000000000000002 < (/.f64 (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 93.4%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 7.5

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

   5. Applied rewrites 7.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 7.5

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

   7. Applied rewrites 7.5%

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

   9. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.04:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (* (sin ky) (/ (sin th) (sqrt (fma -0.5 (cos (* ky -2.0)) 0.5))))
     (if (<= t_1 0.05)
       (* (sin ky) (/ (sin th) (sin kx)))
       (/ 1.0 (/ (sin ky) (* (sin ky) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(-0.5, cos((ky * -2.0)), 0.5)));
	} else if (t_1 <= 0.05) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = 1.0 / (sin(ky) / (sin(ky) * sin(th)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(ky * -2.0)), 0.5))));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = Float64(1.0 / Float64(sin(ky) / Float64(sin(ky) * sin(th))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 88.2%

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 38.5

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

   7. Applied rewrites 38.5%

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

   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))))) < 0.050000000000000003

   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

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

      1. lower-sin.f64 69.1

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

   5. Applied rewrites 69.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 69.2

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

   7. Applied rewrites 69.2%

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

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

   1. Initial program 93.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 7.2

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

   5. Applied rewrites 7.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 7.2

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

   7. Applied rewrites 7.2%

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

   9. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{ky}{\sin kx}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ ky (sin kx))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 0.01) t_1 (if (<= t_2 2.0) (sin th) t_1))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (ky / sin(kx));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 0.01) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(th) * (ky / sin(kx))
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_2 <= 0.01d0) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) * (ky / Math.sin(kx));
	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 <= 0.01) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) * (ky / math.sin(kx))
	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 <= 0.01:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(ky / sin(kx)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 0.01)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) * (ky / sin(kx));
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= 0.01)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.01], t$95$1, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], t$95$1]]]]

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))))) < 0.0100000000000000002 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.0%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 39.7

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

   5. Applied rewrites 39.7%

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

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 98.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 64.8

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

   5. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot \frac{\sin th}{kx}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) (/ (sin th) kx)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 0.01) t_1 (if (<= t_2 2.0) (sin th) t_1))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * (sin(th) / kx);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 0.01) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(ky) * (sin(th) / kx)
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_2 <= 0.01d0) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) * (Math.sin(th) / kx);
	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 <= 0.01) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) * (math.sin(th) / kx)
	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 <= 0.01:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * Float64(sin(th) / kx))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 0.01)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) * (sin(th) / kx);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= 0.01)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $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, 0.01], t$95$1, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], t$95$1]]]]

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))))) < 0.0100000000000000002 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in kx around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 25.7

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

   8. Applied rewrites 25.7%

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

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 98.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 64.8

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

   5. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 34.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{ky \cdot \sin th}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot \left(kx \cdot kx\right), kx\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (/ (* ky (sin th)) (fma -0.16666666666666666 (* kx (* kx kx)) kx)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 0.01) t_1 (if (<= t_2 2.0) (sin th) t_1))))

C

double code(double kx, double ky, double th) {
	double t_1 = (ky * sin(th)) / fma(-0.16666666666666666, (kx * (kx * kx)), kx);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 0.01) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(ky * sin(th)) / fma(-0.16666666666666666, Float64(kx * Float64(kx * kx)), kx))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 0.01)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(-0.16666666666666666 * N[(kx * N[(kx * kx), $MachinePrecision]), $MachinePrecision] + kx), $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, 0.01], t$95$1, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], t$95$1]]]]

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))))) < 0.0100000000000000002 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 38.7

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

   5. Applied rewrites 38.7%

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

   6. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. pow-plus N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{ky \cdot \sin th}{\frac{-1}{6} \cdot {kx}^{\color{blue}{3}} + 1 \cdot kx} \]

      6. cube-unmult N/A

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

      7. unpow2 N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 24.3

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

   8. Applied rewrites 24.3%

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

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 98.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 64.8

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

   5. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 34.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{ky \cdot \sin th}{kx}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* ky (sin th)) kx))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 0.01) t_1 (if (<= t_2 2.0) (sin th) t_1))))

C

double code(double kx, double ky, double th) {
	double t_1 = (ky * sin(th)) / kx;
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 0.01) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	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 * sin(th)) / kx
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_2 <= 0.01d0) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (ky * Math.sin(th)) / kx;
	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 <= 0.01) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (ky * math.sin(th)) / kx
	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 <= 0.01:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(ky * sin(th)) / kx)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 0.01)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (ky * sin(th)) / kx;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= 0.01)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / kx), $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, 0.01], t$95$1, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], t$95$1]]]]

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))))) < 0.0100000000000000002 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 38.7

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

   5. Applied rewrites 38.7%

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

   6. Taylor expanded in kx 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 24.4

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

   8. Applied rewrites 24.4%

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

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 98.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 64.8

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

   5. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 15.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-283}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      2e-283)
   (* -0.16666666666666666 (* th (* th th)))
   (fma
    th
    (* (* th th) (fma 0.008333333333333333 (* th th) -0.16666666666666666))
    th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(th) * (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 2e-283) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(th) * Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 2e-283)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-283], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 1.99999999999999989e-283

   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 19.2

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

   5. Applied rewrites 19.2%

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

   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 9.4

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Applied rewrites 9.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   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 16.7

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Applied rewrites 16.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 1.99999999999999989e-283 < (*.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 90.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 20.9

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

   5. Applied rewrites 20.9%

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

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + \color{blue}{th} \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{th \cdot th}, \frac{-1}{6}\right), th\right) \]

      12. lower-*.f64 11.5

          \[\leadsto \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{th \cdot th}, -0.16666666666666666\right), th\right) \]

   8. Applied rewrites 11.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-283}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 5e-11)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx))))
   (*
    (sin th)
    (*
     (sin ky)
     (sqrt
      (/
       1.0
       (fma (- 1.0 (cos (+ kx kx))) 0.5 (fma (cos (+ ky ky)) -0.5 0.5))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 5e-11) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((1.0 / fma((1.0 - cos((kx + kx))), 0.5, fma(cos((ky + ky)), -0.5, 0.5)))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 5e-11)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(1.0 / fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, fma(cos(Float64(ky + ky)), -0.5, 0.5))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 5e-11], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.00000000000000018e-11

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 5.00000000000000018e-11 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.3%

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. flip-+ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. flip-+ N/A

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

      16. lift-+.f64 N/A

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

      17. lower-fma.f64 29.0

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

   6. Applied rewrites 99.2%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, \mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 5e-11)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx))))
   (*
    (sin ky)
    (/
     (sin th)
     (sqrt
      (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* (cos (+ ky ky)) -0.5))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 5e-11) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	} else {
		tmp = sin(ky) * (sin(th) / sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (cos((ky + ky)) * -0.5)))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 5e-11)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(cos(Float64(ky + ky)) * -0.5))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 5e-11], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.00000000000000018e-11

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 5.00000000000000018e-11 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.3%

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

   4. Applied rewrites 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 5e-11)
   (*
    (sin th)
    (/
     (sin ky)
     (hypot (sin ky) (fma kx (* -0.16666666666666666 (* kx kx)) kx))))
   (*
    (sin th)
    (/
     (sin ky)
     (sqrt
      (fma (- 1.0 (cos (+ kx kx))) 0.5 (+ 0.5 (* (cos (+ ky ky)) -0.5))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 5e-11) {
		tmp = sin(th) * (sin(ky) / hypot(sin(ky), fma(kx, (-0.16666666666666666 * (kx * kx)), kx)));
	} else {
		tmp = sin(th) * (sin(ky) / sqrt(fma((1.0 - cos((kx + kx))), 0.5, (0.5 + (cos((ky + ky)) * -0.5)))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 5e-11)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), fma(kx, Float64(-0.16666666666666666 * Float64(kx * kx)), kx))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(kx + kx))), 0.5, Float64(0.5 + Float64(cos(Float64(ky + ky)) * -0.5))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 5e-11], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(kx * N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] + kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 + N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.00000000000000018e-11

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 5.00000000000000018e-11 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.3%

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

      1. lift-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. lift-+.f64 N/A

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

      6. lift-sqrt.f64 99.3

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. sin-mult N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(kx, -0.16666666666666666 \cdot \left(kx \cdot kx\right), kx\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + \cos \left(ky + ky\right) \cdot -0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-10}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-10)
   (* ky (/ th (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 1e-10) {
		tmp = ky * (th / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 1d-10) then
        tmp = ky * (th / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 1e-10) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 1e-10:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-10)
		tmp = Float64(ky * Float64(th / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-10)
		tmp = ky * (th / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-10], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000004e-10

   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 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 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in th around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 24.3

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

   8. Applied rewrites 24.3%

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

   if 1.00000000000000004e-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 93.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.8

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

   5. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 49.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot \sin th\\ \mathbf{if}\;\sin kx \leq -0.08:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{t\_1}}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) (sin th))))
   (if (<= (sin kx) -0.08)
     (*
      (sin th)
      (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0))))
     (if (<= (sin kx) 5e-165)
       (/ 1.0 (/ (sin ky) t_1))
       (if (<= (sin kx) 4e-75)
         (* (sin th) (/ (sin ky) (sqrt (fma ky ky (* kx kx)))))
         (/ t_1 (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * sin(th);
	double tmp;
	if (sin(kx) <= -0.08) {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	} else if (sin(kx) <= 5e-165) {
		tmp = 1.0 / (sin(ky) / t_1);
	} else if (sin(kx) <= 4e-75) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = t_1 / sin(kx);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * sin(th))
	tmp = 0.0
	if (sin(kx) <= -0.08)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	elseif (sin(kx) <= 5e-165)
		tmp = Float64(1.0 / Float64(sin(ky) / t_1));
	elseif (sin(kx) <= 4e-75)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(t_1 / sin(kx));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.08], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-165], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-75], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[Sin[kx], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0800000000000000017

   1. Initial program 99.3%

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

   4. Applied rewrites 40.0%

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

   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 34.6

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

   7. Applied rewrites 34.6%

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

   if -0.0800000000000000017 < (sin.f64 kx) < 4.99999999999999981e-165

   1. Initial program 83.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 13.4

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

   5. Applied rewrites 13.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 13.4

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

   7. Applied rewrites 13.4%

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

   9. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}} \]

   if 4.99999999999999981e-165 < (sin.f64 kx) < 3.9999999999999998e-75

   1. Initial program 99.3%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.2

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

   8. Applied rewrites 59.2%

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

   if 3.9999999999999998e-75 < (sin.f64 kx)

   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

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

      1. lower-sin.f64 65.5

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

   5. Applied rewrites 65.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 65.5

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

   7. Applied rewrites 65.5%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.08:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin ky \cdot \sin th}}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 49.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot \sin th\\ \mathbf{if}\;\sin kx \leq -0.08:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-165}:\\ \;\;\;\;t\_1 \cdot \frac{1}{\sin ky}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) (sin th))))
   (if (<= (sin kx) -0.08)
     (*
      (sin th)
      (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0))))
     (if (<= (sin kx) 5e-165)
       (* t_1 (/ 1.0 (sin ky)))
       (if (<= (sin kx) 4e-75)
         (* (sin th) (/ (sin ky) (sqrt (fma ky ky (* kx kx)))))
         (/ t_1 (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * sin(th);
	double tmp;
	if (sin(kx) <= -0.08) {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	} else if (sin(kx) <= 5e-165) {
		tmp = t_1 * (1.0 / sin(ky));
	} else if (sin(kx) <= 4e-75) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = t_1 / sin(kx);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * sin(th))
	tmp = 0.0
	if (sin(kx) <= -0.08)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	elseif (sin(kx) <= 5e-165)
		tmp = Float64(t_1 * Float64(1.0 / sin(ky)));
	elseif (sin(kx) <= 4e-75)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(t_1 / sin(kx));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[kx], $MachinePrecision], -0.08], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-165], N[(t$95$1 * N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-75], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[Sin[kx], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0800000000000000017

   1. Initial program 99.3%

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

   4. Applied rewrites 40.0%

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

   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 34.6

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

   7. Applied rewrites 34.6%

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

   if -0.0800000000000000017 < (sin.f64 kx) < 4.99999999999999981e-165

   1. Initial program 83.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 13.4

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

   5. Applied rewrites 13.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 13.4

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

   7. Applied rewrites 13.4%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift-sin.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*l* N/A

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

   9. Applied rewrites 47.6%

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

   if 4.99999999999999981e-165 < (sin.f64 kx) < 3.9999999999999998e-75

   1. Initial program 99.3%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} + {kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{ky \cdot ky} + {kx}^{2}}} \cdot \sin th \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(ky, ky, {kx}^{2}\right)}}} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      5. lower-*.f64 59.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Applied rewrites 59.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}} \cdot \sin th \]

   if 3.9999999999999998e-75 < (sin.f64 kx)

   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

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 65.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 65.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky}}{\sin kx} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \color{blue}{\sin th} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      6. lower-*.f64 65.5

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sin kx} \]

   7. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.08:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\sin ky}\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 49.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.072:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.072)
   (* (sin th) (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0))))
   (if (<= (sin kx) 5e-165)
     (sin th)
     (if (<= (sin kx) 4e-75)
       (* (sin th) (/ (sin ky) (sqrt (fma ky ky (* kx kx)))))
       (/ (* (sin ky) (sin th)) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.072) {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	} else if (sin(kx) <= 5e-165) {
		tmp = sin(th);
	} else if (sin(kx) <= 4e-75) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = (sin(ky) * sin(th)) / sin(kx);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.072)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	elseif (sin(kx) <= 5e-165)
		tmp = sin(th);
	elseif (sin(kx) <= 4e-75)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(Float64(sin(ky) * sin(th)) / sin(kx));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.072], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-165], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-75], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0719999999999999946

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 38.6%

      \[\leadsto \color{blue}{\left(\sqrt{\sin ky} \cdot \sqrt{\frac{\sin ky}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)} \cdot \sin th \]

   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 35.3

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   if -0.0719999999999999946 < (sin.f64 kx) < 4.99999999999999981e-165

   1. Initial program 83.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 41.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\sin th} \]

   if 4.99999999999999981e-165 < (sin.f64 kx) < 3.9999999999999998e-75

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 59.2

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   5. Applied rewrites 59.2%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} + {kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{ky \cdot ky} + {kx}^{2}}} \cdot \sin th \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(ky, ky, {kx}^{2}\right)}}} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      5. lower-*.f64 59.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Applied rewrites 59.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}} \cdot \sin th \]

   if 3.9999999999999998e-75 < (sin.f64 kx)

   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

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 65.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 65.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky}}{\sin kx} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \color{blue}{\sin th} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      6. lower-*.f64 65.5

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sin kx} \]

   7. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.072:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 49.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.072:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.072)
   (* (sin th) (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0))))
   (if (<= (sin kx) 5e-165)
     (sin th)
     (if (<= (sin kx) 4e-75)
       (* (sin th) (/ (sin ky) (sqrt (fma ky ky (* kx kx)))))
       (* (sin ky) (/ (sin th) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.072) {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	} else if (sin(kx) <= 5e-165) {
		tmp = sin(th);
	} else if (sin(kx) <= 4e-75) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = sin(ky) * (sin(th) / sin(kx));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.072)
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	elseif (sin(kx) <= 5e-165)
		tmp = sin(th);
	elseif (sin(kx) <= 4e-75)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.072], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-165], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 4e-75], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0719999999999999946

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 38.6%

      \[\leadsto \color{blue}{\left(\sqrt{\sin ky} \cdot \sqrt{\frac{\sin ky}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)} \cdot \sin th \]

   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 35.3

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]

   if -0.0719999999999999946 < (sin.f64 kx) < 4.99999999999999981e-165

   1. Initial program 83.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 41.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\sin th} \]

   if 4.99999999999999981e-165 < (sin.f64 kx) < 3.9999999999999998e-75

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 59.2

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   5. Applied rewrites 59.2%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} + {kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{ky \cdot ky} + {kx}^{2}}} \cdot \sin th \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(ky, ky, {kx}^{2}\right)}}} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      5. lower-*.f64 59.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Applied rewrites 59.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}} \cdot \sin th \]

   if 3.9999999999999998e-75 < (sin.f64 kx)

   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

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 65.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 65.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky}}{\sin kx} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \color{blue}{\sin th} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      7. lower-/.f64 65.5

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   7. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.072:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 31.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 8 \cdot 10^{-52}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 8e-52)
   (* -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)))) <= 8e-52) {
		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)))) <= 8d-52) 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)))) <= 8e-52) {
		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)))) <= 8e-52:
		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)))) <= 8e-52)
		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)))) <= 8e-52)
		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], 8e-52], 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))))) < 8.0000000000000001e-52

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.6%

      \[\leadsto \color{blue}{\sin th} \]

   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. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 3.4

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   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 14.7

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Applied rewrites 14.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 8.0000000000000001e-52 < (/.f64 (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 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 58.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 36.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6.8 \cdot 10^{-162}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 2000000000:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 6.8e-162)
   (sin th)
   (if (<= kx 2000000000.0)
     (* (sin th) (/ (sin ky) (sqrt (fma ky ky (* kx kx)))))
     (*
      (sin th)
      (* (sqrt (/ 1.0 (- 1.0 (cos (* kx -2.0))))) (* ky (sqrt 2.0)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.8e-162) {
		tmp = sin(th);
	} else if (kx <= 2000000000.0) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = sin(th) * (sqrt((1.0 / (1.0 - cos((kx * -2.0))))) * (ky * sqrt(2.0)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 6.8e-162)
		tmp = sin(th);
	elseif (kx <= 2000000000.0)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(sin(th) * Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(kx * -2.0))))) * Float64(ky * sqrt(2.0))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 6.8e-162], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 2000000000.0], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(ky * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 6.8e-162

   1. Initial program 90.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 27.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\sin th} \]

   if 6.8e-162 < kx < 2e9

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

      2. lower-*.f64 61.0

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   5. Applied rewrites 61.0%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} + {kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{ky \cdot ky} + {kx}^{2}}} \cdot \sin th \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(ky, ky, {kx}^{2}\right)}}} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

      5. lower-*.f64 60.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   8. Applied rewrites 60.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}} \cdot \sin th \]

   if 2e9 < kx

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   4. Applied rewrites 38.7%

      \[\leadsto \color{blue}{\left(\sqrt{\sin ky} \cdot \sqrt{\frac{\sin ky}{\mathsf{fma}\left(1 - \cos \left(kx + kx\right), 0.5, 0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)}}\right)} \cdot \sin th \]

   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 46.5

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sin th \]

   7. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 6.8 \cdot 10^{-162}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 2000000000:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sqrt{\frac{1}{1 - \cos \left(kx \cdot -2\right)}} \cdot \left(ky \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 19.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 8.2 \cdot 10^{-165}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 8.2e-165)
   (* -0.16666666666666666 (* th (* th th)))
   (fma th (* -0.16666666666666666 (* th th)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 8.2e-165) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 8.2e-165)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 8.2e-165], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 8.2000000000000004e-165

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 4.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 4.0%

      \[\leadsto \color{blue}{\sin th} \]

   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. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 3.3

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   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 13.7

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Applied rewrites 13.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 8.2000000000000004e-165 < (sin.f64 ky)

   1. Initial program 99.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 49.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\sin th} \]

   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. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 23.3

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 11.0% accurate, 39.5× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* -0.16666666666666666 (* th (* th th))))

C

double code(double kx, double ky, double th) {
	return -0.16666666666666666 * (th * (th * th));
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (-0.16666666666666666d0) * (th * (th * th))
end function

Java

public static double code(double kx, double ky, double th) {
	return -0.16666666666666666 * (th * (th * th));
}

Python

def code(kx, ky, th):
	return -0.16666666666666666 * (th * (th * th))

Julia

function code(kx, ky, th)
	return Float64(-0.16666666666666666 * Float64(th * Float64(th * th)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = -0.16666666666666666 * (th * (th * th));
end

Wolfram

code[kx_, ky_, th_] := N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 19.9

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 19.9%

   \[\leadsto \color{blue}{\sin th} \]

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. +-commutative N/A

      \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   7. lower-*.f64 10.2

      \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

8. Applied rewrites 10.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

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 11.3

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

11. Applied rewrites 11.3%

    \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))