Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.9% → 99.7%

Time: 11.9s

Alternatives: 22

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 83.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 5 regimes

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

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.9

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   9. Applied rewrites 3.2%

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

   11. Applied rewrites 80.2%

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

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

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.3

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 19.9

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-+l- N/A

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

      14. lower--.f64 N/A

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

   10. Applied rewrites 45.8%

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

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

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.2

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

   10. Applied rewrites 98.2%

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

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

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.2

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 55.1%

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

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

   1. Initial program 95.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.7%

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

Alternative 3: 82.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 5 regimes

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

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.9

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   9. Applied rewrites 3.2%

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

   11. Applied rewrites 80.2%

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

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

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.3

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 19.9

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-+l- N/A

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

      14. lower--.f64 N/A

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

   10. Applied rewrites 45.8%

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

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

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.2

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

   10. Applied rewrites 98.2%

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

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

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.2

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 55.6

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

   7. Applied rewrites 55.6%

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

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

   1. Initial program 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{{\sin ky}^{2}}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.001:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{1 - \left(\cos \left(2 \cdot ky\right) - \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999996186923419:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = cos(Float64(2.0 * ky))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(Float64(sin(th) * sin(ky)) / sqrt(Float64(Float64(1.0 - t_2) * 0.5)));
	elseif (t_1 <= -0.001)
		tmp = Float64(Float64(th * sin(ky)) * sqrt(Float64(2.0 / Float64(1.0 - Float64(t_2 - Float64(1.0 - cos(Float64(2.0 * kx))))))));
	elseif (t_1 <= 0.05)
		tmp = Float64(Float64(Float64(fma(fma(Float64(ky * ky), 0.008333333333333333, -0.16666666666666666), Float64(ky * ky), 1.0) * ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	elseif (t_1 <= 0.9999996186923419)
		tmp = Float64(Float64(fma(fma(Float64(th * th), 0.008333333333333333, -0.16666666666666666), Float64(th * th), 1.0) * th) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5))) / sin(ky)));
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(0.5 * kx), Float64(kx / ky), 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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.001], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[(t$95$2 - N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999996186923419], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(0.5 * kx), $MachinePrecision] * N[(kx / ky), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.9

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   9. Applied rewrites 3.2%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. metadata-eval N/A

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

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

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

       5. lower--.f64 N/A

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

       6. lower-cos.f64 N/A

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

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

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 63.2

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

   12. Applied rewrites 63.2%

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

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

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.3

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 19.9

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-+l- N/A

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

      14. lower--.f64 N/A

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

   10. Applied rewrites 45.8%

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

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

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.2

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

   10. Applied rewrites 98.2%

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

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

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.2

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 55.6

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

   7. Applied rewrites 55.6%

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

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

   1. Initial program 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.001:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{1 - \left(\cos \left(2 \cdot ky\right) - \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999996186923419:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(2 \cdot ky\right)\\ t_2 := 1 - t\_1\\ t_3 := th \cdot \sin ky\\ t_4 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_5 := \cos \left(2 \cdot kx\right)\\ \mathbf{if}\;t\_4 \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{t\_2 \cdot 0.5}}\\ \mathbf{elif}\;t\_4 \leq -0.001:\\ \;\;\;\;t\_3 \cdot \sqrt{\frac{2}{1 - \left(t\_1 - \left(1 - t\_5\right)\right)}}\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9999996186923419:\\ \;\;\;\;\sqrt{\frac{2}{1 - \left(t\_5 - t\_2\right)}} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* 2.0 ky)))
        (t_2 (- 1.0 t_1))
        (t_3 (* th (sin ky)))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_5 (cos (* 2.0 kx))))
   (if (<= t_4 -0.98)
     (/ (* (sin th) (sin ky)) (sqrt (* t_2 0.5)))
     (if (<= t_4 -0.001)
       (* t_3 (sqrt (/ 2.0 (- 1.0 (- t_1 (- 1.0 t_5))))))
       (if (<= t_4 0.05)
         (*
          (/
           (*
            (fma
             (fma (* ky ky) 0.008333333333333333 -0.16666666666666666)
             (* ky ky)
             1.0)
            ky)
           (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
          (sin th))
         (if (<= t_4 0.9999996186923419)
           (* (sqrt (/ 2.0 (- 1.0 (- t_5 t_2)))) t_3)
           (* (/ (sin ky) (fma (* 0.5 kx) (/ kx ky) (sin ky))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((2.0 * ky));
	double t_2 = 1.0 - t_1;
	double t_3 = th * sin(ky);
	double t_4 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_5 = cos((2.0 * kx));
	double tmp;
	if (t_4 <= -0.98) {
		tmp = (sin(th) * sin(ky)) / sqrt((t_2 * 0.5));
	} else if (t_4 <= -0.001) {
		tmp = t_3 * sqrt((2.0 / (1.0 - (t_1 - (1.0 - t_5)))));
	} else if (t_4 <= 0.05) {
		tmp = ((fma(fma((ky * ky), 0.008333333333333333, -0.16666666666666666), (ky * ky), 1.0) * ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else if (t_4 <= 0.9999996186923419) {
		tmp = sqrt((2.0 / (1.0 - (t_5 - t_2)))) * t_3;
	} else {
		tmp = (sin(ky) / fma((0.5 * kx), (kx / ky), sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(2.0 * ky))
	t_2 = Float64(1.0 - t_1)
	t_3 = Float64(th * sin(ky))
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_5 = cos(Float64(2.0 * kx))
	tmp = 0.0
	if (t_4 <= -0.98)
		tmp = Float64(Float64(sin(th) * sin(ky)) / sqrt(Float64(t_2 * 0.5)));
	elseif (t_4 <= -0.001)
		tmp = Float64(t_3 * sqrt(Float64(2.0 / Float64(1.0 - Float64(t_1 - Float64(1.0 - t_5))))));
	elseif (t_4 <= 0.05)
		tmp = Float64(Float64(Float64(fma(fma(Float64(ky * ky), 0.008333333333333333, -0.16666666666666666), Float64(ky * ky), 1.0) * ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	elseif (t_4 <= 0.9999996186923419)
		tmp = Float64(sqrt(Float64(2.0 / Float64(1.0 - Float64(t_5 - t_2)))) * t_3);
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(0.5 * kx), Float64(kx / ky), sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -0.98], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.001], N[(t$95$3 * N[Sqrt[N[(2.0 / N[(1.0 - N[(t$95$1 - N[(1.0 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.05], N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9999996186923419], N[(N[Sqrt[N[(2.0 / N[(1.0 - N[(t$95$5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(0.5 * kx), $MachinePrecision] * N[(kx / ky), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.9

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   9. Applied rewrites 3.2%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. metadata-eval N/A

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

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

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

       5. lower--.f64 N/A

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

       6. lower-cos.f64 N/A

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

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

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 63.2

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

   12. Applied rewrites 63.2%

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

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

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.3

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 19.9

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-+l- N/A

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

      14. lower--.f64 N/A

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

   10. Applied rewrites 45.8%

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

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

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.2

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

   10. Applied rewrites 98.2%

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

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

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.2

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 20.2

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

   7. Applied rewrites 20.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 20.2

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

   9. Applied rewrites 21.4%

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

   10. Taylor expanded in th around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. distribute-lft-out N/A

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

       6. associate-/r* N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 N/A

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

       9. metadata-eval N/A

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

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

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

       11. associate-+l- N/A

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

       12. lower--.f64 N/A

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

   12. Applied rewrites 56.2%

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

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

   1. Initial program 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.001:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{1 - \left(\cos \left(2 \cdot ky\right) - \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.008333333333333333, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999996186923419:\\ \;\;\;\;\sqrt{\frac{2}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(2 \cdot ky\right)\\ t_2 := 1 - t\_1\\ t_3 := th \cdot \sin ky\\ t_4 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_5 := \cos \left(2 \cdot kx\right)\\ \mathbf{if}\;t\_4 \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{t\_2 \cdot 0.5}}\\ \mathbf{elif}\;t\_4 \leq -0.001:\\ \;\;\;\;t\_3 \cdot \sqrt{\frac{2}{1 - \left(t\_1 - \left(1 - t\_5\right)\right)}}\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9999996186923419:\\ \;\;\;\;\sqrt{\frac{2}{1 - \left(t\_5 - t\_2\right)}} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* 2.0 ky)))
        (t_2 (- 1.0 t_1))
        (t_3 (* th (sin ky)))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_5 (cos (* 2.0 kx))))
   (if (<= t_4 -0.98)
     (/ (* (sin th) (sin ky)) (sqrt (* t_2 0.5)))
     (if (<= t_4 -0.001)
       (* t_3 (sqrt (/ 2.0 (- 1.0 (- t_1 (- 1.0 t_5))))))
       (if (<= t_4 0.05)
         (*
          (/
           (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
           (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
          (sin th))
         (if (<= t_4 0.9999996186923419)
           (* (sqrt (/ 2.0 (- 1.0 (- t_5 t_2)))) t_3)
           (* (/ (sin ky) (fma (* 0.5 kx) (/ kx ky) (sin ky))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((2.0 * ky));
	double t_2 = 1.0 - t_1;
	double t_3 = th * sin(ky);
	double t_4 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_5 = cos((2.0 * kx));
	double tmp;
	if (t_4 <= -0.98) {
		tmp = (sin(th) * sin(ky)) / sqrt((t_2 * 0.5));
	} else if (t_4 <= -0.001) {
		tmp = t_3 * sqrt((2.0 / (1.0 - (t_1 - (1.0 - t_5)))));
	} else if (t_4 <= 0.05) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else if (t_4 <= 0.9999996186923419) {
		tmp = sqrt((2.0 / (1.0 - (t_5 - t_2)))) * t_3;
	} else {
		tmp = (sin(ky) / fma((0.5 * kx), (kx / ky), sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(2.0 * ky))
	t_2 = Float64(1.0 - t_1)
	t_3 = Float64(th * sin(ky))
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_5 = cos(Float64(2.0 * kx))
	tmp = 0.0
	if (t_4 <= -0.98)
		tmp = Float64(Float64(sin(th) * sin(ky)) / sqrt(Float64(t_2 * 0.5)));
	elseif (t_4 <= -0.001)
		tmp = Float64(t_3 * sqrt(Float64(2.0 / Float64(1.0 - Float64(t_1 - Float64(1.0 - t_5))))));
	elseif (t_4 <= 0.05)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	elseif (t_4 <= 0.9999996186923419)
		tmp = Float64(sqrt(Float64(2.0 / Float64(1.0 - Float64(t_5 - t_2)))) * t_3);
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(0.5 * kx), Float64(kx / ky), sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -0.98], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.001], N[(t$95$3 * N[Sqrt[N[(2.0 / N[(1.0 - N[(t$95$1 - N[(1.0 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.05], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9999996186923419], N[(N[Sqrt[N[(2.0 / N[(1.0 - N[(t$95$5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(0.5 * kx), $MachinePrecision] * N[(kx / ky), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.9

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 63.2

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

   9. Applied rewrites 3.2%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. metadata-eval N/A

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

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

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

       5. lower--.f64 N/A

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

       6. lower-cos.f64 N/A

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

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

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 63.2

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

   12. Applied rewrites 63.2%

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

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

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.3

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 19.9

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-+l- N/A

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

      14. lower--.f64 N/A

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

   10. Applied rewrites 45.8%

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

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

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 98.2

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 98.2%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 97.9

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   10. Applied rewrites 97.9%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999961869234189

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 99.2

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 20.2

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 20.2%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      8. lower-*.f64 20.2

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}} \]

   9. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}} \]

   10. Taylor expanded in th around 0

       \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(th \cdot \sin ky\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

       3. lower-sin.f64 N/A

          \[\leadsto \left(th \cdot \color{blue}{\sin ky}\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

       4. lower-sqrt.f64 N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

       5. distribute-lft-out N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{2} \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

       6. associate-/r* N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

       7. metadata-eval N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{\color{blue}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

       8. lower-/.f64 N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{2}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

       9. metadata-eval N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

       10. distribute-lft-neg-in N/A

           \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot kx\right)\right)}\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

       11. associate-+l- N/A

           \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{\color{blue}{1 - \left(\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

       12. lower--.f64 N/A

           \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{\color{blue}{1 - \left(\cos \left(\mathsf{neg}\left(-2 \cdot kx\right)\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   12. Applied rewrites 56.2%

       \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{1 - \left(\cos \left(kx \cdot 2\right) - \left(1 - \cos \left(ky \cdot 2\right)\right)\right)}}} \]

   if 0.99999961869234189 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 95.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\frac{1}{2} \cdot {kx}^{2}}}{\sin ky} + \sin ky} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\frac{1}{2} \cdot \color{blue}{\left(kx \cdot kx\right)}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\left(\frac{1}{2} \cdot kx\right) \cdot kx}}{\sin ky} + \sin ky} \cdot \sin th \]

      7. associate-/l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(\frac{1}{2} \cdot kx\right) \cdot \frac{kx}{\sin ky}} + \sin ky} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot kx}, \frac{kx}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \color{blue}{\frac{kx}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. lower-sin.f64 100.0

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.001:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{1 - \left(\cos \left(2 \cdot ky\right) - \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999996186923419:\\ \;\;\;\;\sqrt{\frac{2}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}} \cdot \left(th \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \cos \left(2 \cdot ky\right)\\ t_3 := \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{1 - \left(t\_2 - \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - t\_2\right) \cdot 0.5}}\\ \mathbf{elif}\;t\_1 \leq -0.001:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.9999996186923419:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (cos (* 2.0 ky)))
        (t_3
         (*
          (* th (sin ky))
          (sqrt (/ 2.0 (- 1.0 (- t_2 (- 1.0 (cos (* 2.0 kx))))))))))
   (if (<= t_1 -0.98)
     (/ (* (sin th) (sin ky)) (sqrt (* (- 1.0 t_2) 0.5)))
     (if (<= t_1 -0.001)
       t_3
       (if (<= t_1 0.05)
         (*
          (/
           (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
           (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
          (sin th))
         (if (<= t_1 0.9999996186923419)
           t_3
           (* (/ (sin ky) (fma (* 0.5 kx) (/ kx ky) (sin ky))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = cos((2.0 * ky));
	double t_3 = (th * sin(ky)) * sqrt((2.0 / (1.0 - (t_2 - (1.0 - cos((2.0 * kx)))))));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = (sin(th) * sin(ky)) / sqrt(((1.0 - t_2) * 0.5));
	} else if (t_1 <= -0.001) {
		tmp = t_3;
	} else if (t_1 <= 0.05) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else if (t_1 <= 0.9999996186923419) {
		tmp = t_3;
	} else {
		tmp = (sin(ky) / fma((0.5 * kx), (kx / ky), sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = cos(Float64(2.0 * ky))
	t_3 = Float64(Float64(th * sin(ky)) * sqrt(Float64(2.0 / Float64(1.0 - Float64(t_2 - Float64(1.0 - cos(Float64(2.0 * kx))))))))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(Float64(sin(th) * sin(ky)) / sqrt(Float64(Float64(1.0 - t_2) * 0.5)));
	elseif (t_1 <= -0.001)
		tmp = t_3;
	elseif (t_1 <= 0.05)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	elseif (t_1 <= 0.9999996186923419)
		tmp = t_3;
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(0.5 * kx), Float64(kx / ky), 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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 / N[(1.0 - N[(t$95$2 - N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.001], t$95$3, If[LessEqual[t$95$1, 0.05], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999996186923419], t$95$3, N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(0.5 * kx), $MachinePrecision] * N[(kx / ky), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $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 89.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 88.9

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 63.4

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 63.4%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      8. lower-*.f64 63.2

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}} \]

   9. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}} \]

   10. Taylor expanded in kx around 0

       \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}} \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}} \]

       9. *-commutative N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}} \]

       10. lower-*.f64 63.2

           \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}} \]

   12. Applied rewrites 63.2%

       \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}} \]

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3 or 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999961869234189

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 99.3

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 20.0

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 20.0%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      6. distribute-lft-out N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{2} \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      7. associate-/r* N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      8. metadata-eval N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{\color{blue}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{2}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      10. +-commutative N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}} \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}} \]

      13. associate-+l- N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{\color{blue}{1 - \left(\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right) - \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}} \]

      14. lower--.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{\color{blue}{1 - \left(\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right) - \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}} \]

   10. Applied rewrites 50.7%

       \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{2}{1 - \left(\cos \left(ky \cdot 2\right) - \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}} \]

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 98.2

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 98.2%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 97.9

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   10. Applied rewrites 97.9%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   if 0.99999961869234189 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 95.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\frac{1}{2} \cdot {kx}^{2}}}{\sin ky} + \sin ky} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\frac{1}{2} \cdot \color{blue}{\left(kx \cdot kx\right)}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\left(\frac{1}{2} \cdot kx\right) \cdot kx}}{\sin ky} + \sin ky} \cdot \sin th \]

      7. associate-/l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(\frac{1}{2} \cdot kx\right) \cdot \frac{kx}{\sin ky}} + \sin ky} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot kx}, \frac{kx}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \color{blue}{\frac{kx}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. lower-sin.f64 100.0

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.001:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{1 - \left(\cos \left(2 \cdot ky\right) - \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999996186923419:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{2}{1 - \left(\cos \left(2 \cdot ky\right) - \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.708:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \mathbf{elif}\;t\_1 \leq 0.999:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.708)
     (/ (* (sin th) (sin ky)) (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5)))
     (if (<= t_1 0.999)
       (* (/ (sin th) (sqrt (* (- 1.0 (cos (+ kx kx))) 0.5))) (sin ky))
       (* (/ (sin ky) (fma (* 0.5 kx) (/ kx ky) (sin ky))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.708) {
		tmp = (sin(th) * sin(ky)) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5));
	} else if (t_1 <= 0.999) {
		tmp = (sin(th) / sqrt(((1.0 - cos((kx + kx))) * 0.5))) * sin(ky);
	} else {
		tmp = (sin(ky) / fma((0.5 * kx), (kx / ky), sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.708)
		tmp = Float64(Float64(sin(th) * sin(ky)) / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5)));
	elseif (t_1 <= 0.999)
		tmp = Float64(Float64(sin(th) / sqrt(Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5))) * sin(ky));
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(0.5 * kx), Float64(kx / ky), 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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.708], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.999], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(0.5 * kx), $MachinePrecision] * N[(kx / ky), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $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))))) < -0.70799999999999996

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 90.8

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 55.8

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 55.8%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      8. lower-*.f64 55.7

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}} \]

   9. Applied rewrites 6.1%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}} \]

   10. Taylor expanded in kx around 0

       \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}} \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}} \]

       9. *-commutative N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}} \]

       10. lower-*.f64 55.7

           \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}} \]

   12. Applied rewrites 55.7%

       \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}} \]

   if -0.70799999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 99.4

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 9.0

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 9.0%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

   9. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky} \]

   if 0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 95.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\frac{1}{2} \cdot {kx}^{2}}}{\sin ky} + \sin ky} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\frac{1}{2} \cdot \color{blue}{\left(kx \cdot kx\right)}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\left(\frac{1}{2} \cdot kx\right) \cdot kx}}{\sin ky} + \sin ky} \cdot \sin th \]

      7. associate-/l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(\frac{1}{2} \cdot kx\right) \cdot \frac{kx}{\sin ky}} + \sin ky} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot kx}, \frac{kx}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \color{blue}{\frac{kx}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. lower-sin.f64 100.0

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.708:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.999:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)\right)}}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 0.999:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.88)
     (/
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/
       (sqrt
        (fma
         (- 1.0 (cos (* 2.0 ky)))
         0.5
         (*
          (fma
           (fma 0.044444444444444446 (* kx kx) -0.3333333333333333)
           (* kx kx)
           1.0)
          (* kx kx))))
       (sin ky)))
     (if (<= t_1 0.999)
       (* (/ (sin th) (sqrt (* (- 1.0 (cos (+ kx kx))) 0.5))) (sin ky))
       (* (/ (sin ky) (fma (* 0.5 kx) (/ kx ky) (sin ky))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.88) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / (sqrt(fma((1.0 - cos((2.0 * ky))), 0.5, (fma(fma(0.044444444444444446, (kx * kx), -0.3333333333333333), (kx * kx), 1.0) * (kx * kx)))) / sin(ky));
	} else if (t_1 <= 0.999) {
		tmp = (sin(th) / sqrt(((1.0 - cos((kx + kx))) * 0.5))) * sin(ky);
	} else {
		tmp = (sin(ky) / fma((0.5 * kx), (kx / ky), sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.88)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 0.5, Float64(fma(fma(0.044444444444444446, Float64(kx * kx), -0.3333333333333333), Float64(kx * kx), 1.0) * Float64(kx * kx)))) / sin(ky)));
	elseif (t_1 <= 0.999)
		tmp = Float64(Float64(sin(th) / sqrt(Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5))) * sin(ky));
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(0.5 * kx), Float64(kx / ky), 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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.88], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(N[(0.044444444444444446 * N[(kx * kx), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.999], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(0.5 * kx), $MachinePrecision] * N[(kx / ky), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $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))))) < -0.880000000000000004

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 89.8

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 59.9

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 59.9%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. lower-*.f64 33.1

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}{\sin ky}} \]

   10. Applied rewrites 33.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}{\sin ky}} \]

   11. Taylor expanded in kx around 0

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right) + {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)}}}{\sin ky}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}} + {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)}}{\sin ky}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}}{\sin ky}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(\color{blue}{2} \cdot ky\right), \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       9. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), \frac{1}{2}, \color{blue}{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot {kx}^{2}}\right)}}{\sin ky}} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), \frac{1}{2}, \color{blue}{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot {kx}^{2}}\right)}}{\sin ky}} \]

   13. Applied rewrites 34.3%

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)\right)}}}{\sin ky}} \]

   if -0.880000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 99.4

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 9.6

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 9.6%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

   9. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky} \]

   if 0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 95.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin ky + \frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{\sin ky} + \sin ky}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2}}{\sin ky} \cdot \frac{1}{2}} + \sin ky} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{kx}^{2} \cdot \frac{1}{2}}{\sin ky}} + \sin ky} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\frac{1}{2} \cdot {kx}^{2}}}{\sin ky} + \sin ky} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\frac{1}{2} \cdot \color{blue}{\left(kx \cdot kx\right)}}{\sin ky} + \sin ky} \cdot \sin th \]

      6. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\left(\frac{1}{2} \cdot kx\right) \cdot kx}}{\sin ky} + \sin ky} \cdot \sin th \]

      7. associate-/l* N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left(\frac{1}{2} \cdot kx\right) \cdot \frac{kx}{\sin ky}} + \sin ky} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot kx}, \frac{kx}{\sin ky}, \sin ky\right)} \cdot \sin th \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \color{blue}{\frac{kx}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\color{blue}{\sin ky}}, \sin ky\right)} \cdot \sin th \]

      12. lower-sin.f64 100.0

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot kx, \frac{kx}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\color{blue}{ky}}, \sin ky\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)\right)}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.999:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)\right)}}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 0.71:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.88)
     (/
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/
       (sqrt
        (fma
         (- 1.0 (cos (* 2.0 ky)))
         0.5
         (*
          (fma
           (fma 0.044444444444444446 (* kx kx) -0.3333333333333333)
           (* kx kx)
           1.0)
          (* kx kx))))
       (sin ky)))
     (if (<= t_1 0.71)
       (* (/ (sin th) (sqrt (* (- 1.0 (cos (+ kx kx))) 0.5))) (sin ky))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.88) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / (sqrt(fma((1.0 - cos((2.0 * ky))), 0.5, (fma(fma(0.044444444444444446, (kx * kx), -0.3333333333333333), (kx * kx), 1.0) * (kx * kx)))) / sin(ky));
	} else if (t_1 <= 0.71) {
		tmp = (sin(th) / sqrt(((1.0 - cos((kx + kx))) * 0.5))) * sin(ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.88)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 0.5, Float64(fma(fma(0.044444444444444446, Float64(kx * kx), -0.3333333333333333), Float64(kx * kx), 1.0) * Float64(kx * kx)))) / sin(ky)));
	elseif (t_1 <= 0.71)
		tmp = Float64(Float64(sin(th) / sqrt(Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5))) * sin(ky));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.88], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(N[(0.044444444444444446 * N[(kx * kx), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.71], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.880000000000000004

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 89.8

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 59.9

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 59.9%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. lower-*.f64 33.1

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}{\sin ky}} \]

   10. Applied rewrites 33.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}{\sin ky}} \]

   11. Taylor expanded in kx around 0

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right) + {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)}}}{\sin ky}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}} + {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)}}{\sin ky}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}}{\sin ky}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(\color{blue}{2} \cdot ky\right), \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       9. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), \frac{1}{2}, \color{blue}{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot {kx}^{2}}\right)}}{\sin ky}} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), \frac{1}{2}, \color{blue}{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot {kx}^{2}}\right)}}{\sin ky}} \]

   13. Applied rewrites 34.3%

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)\right)}}}{\sin ky}} \]

   if -0.880000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.70999999999999996

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 99.5

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 8.4

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 8.4%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

   9. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky} \]

   if 0.70999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 96.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 82.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)\right)}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.71:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)\right)}}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 0.71:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.88)
     (/
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/
       (sqrt
        (fma
         (- 1.0 (cos (* 2.0 ky)))
         0.5
         (*
          (fma
           (fma 0.044444444444444446 (* kx kx) -0.3333333333333333)
           (* kx kx)
           1.0)
          (* kx kx))))
       (sin ky)))
     (if (<= t_1 0.71)
       (* (/ (sin ky) (sqrt (* (- 1.0 (cos (+ kx kx))) 0.5))) (sin th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.88) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / (sqrt(fma((1.0 - cos((2.0 * ky))), 0.5, (fma(fma(0.044444444444444446, (kx * kx), -0.3333333333333333), (kx * kx), 1.0) * (kx * kx)))) / sin(ky));
	} else if (t_1 <= 0.71) {
		tmp = (sin(ky) / sqrt(((1.0 - cos((kx + kx))) * 0.5))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.88)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 0.5, Float64(fma(fma(0.044444444444444446, Float64(kx * kx), -0.3333333333333333), Float64(kx * kx), 1.0) * Float64(kx * kx)))) / sin(ky)));
	elseif (t_1 <= 0.71)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.88], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(N[(0.044444444444444446 * N[(kx * kx), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.71], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.880000000000000004

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 89.8

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 59.9

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 59.9%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. lower-*.f64 33.1

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}{\sin ky}} \]

   10. Applied rewrites 33.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}{\sin ky}} \]

   11. Taylor expanded in kx around 0

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right) + {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)}}}{\sin ky}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}} + {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)}}{\sin ky}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}}{\sin ky}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right), \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(\color{blue}{2} \cdot ky\right), \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       9. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}, \frac{1}{2}, {kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)\right)}}{\sin ky}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), \frac{1}{2}, \color{blue}{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot {kx}^{2}}\right)}}{\sin ky}} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), \frac{1}{2}, \color{blue}{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot {kx}^{2}}\right)}}{\sin ky}} \]

   13. Applied rewrites 34.3%

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)\right)}}}{\sin ky}} \]

   if -0.880000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.70999999999999996

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 99.5

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 8.4

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 8.4%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}{\sin th}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \cdot \sin th} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \cdot \sin th \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin th} \]

   9. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin th} \]

   if 0.70999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 96.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 82.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)\right)}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.71:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{0.5 \cdot \left(1 - \cos \left(ky + ky\right)\right)}}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.02)
     (/
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/ (sqrt (* 0.5 (- 1.0 (cos (+ ky ky))))) (sin ky)))
     (if (<= t_1 0.05)
       (/
        (* (* (fma (* ky ky) -0.16666666666666666 1.0) ky) (sin th))
        (sqrt (* (- 1.0 (cos (+ kx kx))) 0.5)))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / (sqrt((0.5 * (1.0 - cos((ky + ky))))) / sin(ky));
	} else if (t_1 <= 0.05) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) * sin(th)) / sqrt(((1.0 - cos((kx + kx))) * 0.5));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(ky + ky))))) / sin(ky)));
	elseif (t_1 <= 0.05)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) * sin(th)) / sqrt(Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 92.7

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 47.7

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 47.7%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. lower-*.f64 26.4

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}{\sin ky}} \]

   10. Applied rewrites 26.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}{\sin ky}} \]
   11. Step-by-step derivation

   12. Applied rewrites 26.4%

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{\left(1 - \cos \left(ky + ky\right)\right) \cdot 0.5}}{\sin ky}} \]

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 99.5

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 2.6

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 2.6%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      8. lower-*.f64 2.3

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}} \]

   9. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}} \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{\sin th \cdot \color{blue}{\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)\right)}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\sin th \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky\right)}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky\right)}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       3. +-commutative N/A

          \[\leadsto \frac{\sin th \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky\right)}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\sin th \cdot \left(\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky\right)}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th \cdot \left(\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky\right)}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky\right)}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       7. lower-*.f64 74.0

          \[\leadsto \frac{\sin th \cdot \left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky\right)}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \]

   12. Applied rewrites 74.0%

       \[\leadsto \frac{\sin th \cdot \color{blue}{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 97.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{0.5 \cdot \left(1 - \cos \left(ky + ky\right)\right)}}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.001:\\ \;\;\;\;\frac{th \cdot \sin ky}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;\frac{\sin th \cdot ky}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sqrt (* (- 1.0 (cos (+ kx kx))) 0.5)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_2 -0.001)
     (/ (* th (sin ky)) t_1)
     (if (<= t_2 0.05) (/ (* (sin th) ky) t_1) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sqrt(((1.0 - cos((kx + kx))) * 0.5));
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_2 <= -0.001) {
		tmp = (th * sin(ky)) / t_1;
	} else if (t_2 <= 0.05) {
		tmp = (sin(th) * ky) / t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(((1.0d0 - cos((kx + kx))) * 0.5d0))
    t_2 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_2 <= (-0.001d0)) then
        tmp = (th * sin(ky)) / t_1
    else if (t_2 <= 0.05d0) then
        tmp = (sin(th) * ky) / t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sqrt(((1.0 - Math.cos((kx + kx))) * 0.5));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_2 <= -0.001) {
		tmp = (th * Math.sin(ky)) / t_1;
	} else if (t_2 <= 0.05) {
		tmp = (Math.sin(th) * ky) / t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sqrt(((1.0 - math.cos((kx + kx))) * 0.5))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_2 <= -0.001:
		tmp = (th * math.sin(ky)) / t_1
	elif t_2 <= 0.05:
		tmp = (math.sin(th) * ky) / t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sqrt(Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.001)
		tmp = Float64(Float64(th * sin(ky)) / t_1);
	elseif (t_2 <= 0.05)
		tmp = Float64(Float64(sin(th) * ky) / t_1);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sqrt(((1.0 - cos((kx + kx))) * 0.5));
	t_2 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.001)
		tmp = (th * sin(ky)) / t_1;
	elseif (t_2 <= 0.05)
		tmp = (sin(th) * ky) / t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.001], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.05], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1e-3

   1. Initial program 92.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 92.7

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 47.7

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 47.7%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      8. lower-*.f64 47.6

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}} \]

   9. Applied rewrites 10.0%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}} \]

   10. Taylor expanded in th around 0

       \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       2. lower-sin.f64 6.2

          \[\leadsto \frac{th \cdot \color{blue}{\sin ky}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \]

   12. Applied rewrites 6.2%

       \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \]

   if -1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 99.5

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 2.6

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 2.6%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      8. lower-*.f64 2.3

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}} \]

   9. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}} \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{\color{blue}{ky \cdot \sin th}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       3. lower-sin.f64 73.9

          \[\leadsto \frac{\color{blue}{\sin th} \cdot ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \]

   12. Applied rewrites 73.9%

       \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 97.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.001:\\ \;\;\;\;\frac{th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.05)
   (/
    (* (* (fma (* ky ky) -0.16666666666666666 1.0) ky) (sin th))
    (sqrt (* (- 1.0 (cos (+ kx kx))) 0.5)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.05) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) * sin(th)) / sqrt(((1.0 - cos((kx + kx))) * 0.5));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.05)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) * sin(th)) / sqrt(Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $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))))) < 0.050000000000000003

   1. Initial program 95.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 95.9

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 26.6

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 26.6%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      8. lower-*.f64 26.4

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}} \]

   9. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}} \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{\sin th \cdot \color{blue}{\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)\right)}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\sin th \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky\right)}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky\right)}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       3. +-commutative N/A

          \[\leadsto \frac{\sin th \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky\right)}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\sin th \cdot \left(\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky\right)}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\sin th \cdot \left(\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky\right)}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky\right)}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

       7. lower-*.f64 35.6

          \[\leadsto \frac{\sin th \cdot \left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky\right)}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \]

   12. Applied rewrites 35.6%

       \[\leadsto \frac{\sin th \cdot \color{blue}{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)}}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 97.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \sin th}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 2e-8)
   (* (/ (sin th) (sin kx)) ky)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 2e-8) {
		tmp = (sin(th) / sin(kx)) * ky;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 2d-8) then
        tmp = (sin(th) / sin(kx)) * ky
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 2e-8) {
		tmp = (Math.sin(th) / Math.sin(kx)) * ky;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 2e-8:
		tmp = (math.sin(th) / math.sin(kx)) * ky
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 2e-8)
		tmp = Float64(Float64(sin(th) / sin(kx)) * ky);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 2e-8)
		tmp = (sin(th) / sin(kx)) * ky;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right) \cdot ky} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right) \cdot ky} \]

   5. Applied rewrites 23.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{\sin th}{{\sin kx}^{3}}, -0.5, \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \frac{\sin th}{\sin kx}\right) \cdot ky} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\sin kx} \cdot ky \]
   7. Step-by-step derivation

   8. Applied rewrites 25.7%

      \[\leadsto \frac{\sin th}{\sin kx} \cdot ky \]

   if 2e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 64.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-71}:\\ \;\;\;\;\left(\frac{\left(ky \cdot ky\right) \cdot \sin th}{\left(kx \cdot kx\right) \cdot kx} \cdot -0.5\right) \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-71)
   (* (* (/ (* (* ky ky) (sin th)) (* (* kx kx) kx)) -0.5) ky)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-71) {
		tmp = ((((ky * ky) * sin(th)) / ((kx * kx) * kx)) * -0.5) * ky;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-71) then
        tmp = ((((ky * ky) * sin(th)) / ((kx * kx) * kx)) * (-0.5d0)) * ky
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-71) {
		tmp = ((((ky * ky) * Math.sin(th)) / ((kx * kx) * kx)) * -0.5) * ky;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-71:
		tmp = ((((ky * ky) * math.sin(th)) / ((kx * kx) * kx)) * -0.5) * ky
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-71)
		tmp = Float64(Float64(Float64(Float64(Float64(ky * ky) * sin(th)) / Float64(Float64(kx * kx) * kx)) * -0.5) * ky);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-71)
		tmp = ((((ky * ky) * sin(th)) / ((kx * kx) * kx)) * -0.5) * ky;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-71], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(kx * kx), $MachinePrecision] * kx), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * ky), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999992e-72

   1. Initial program 95.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky \cdot \left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right) \cdot ky} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({ky}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sin th}{{\sin kx}^{3}} + \frac{-1}{6} \cdot \frac{\sin th}{\sin kx}\right) + \frac{\sin th}{\sin kx}\right) \cdot ky} \]

   5. Applied rewrites 22.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{\sin th}{{\sin kx}^{3}}, -0.5, \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \frac{\sin th}{\sin kx}\right) \cdot ky} \]

   6. Taylor expanded in kx around 0

      \[\leadsto \left(\frac{-1}{2} \cdot \frac{{ky}^{2} \cdot \sin th}{{kx}^{3}}\right) \cdot ky \]
   7. Step-by-step derivation

   8. Applied rewrites 9.1%

      \[\leadsto \left(\frac{\left(ky \cdot ky\right) \cdot \sin th}{\left(kx \cdot kx\right) \cdot kx} \cdot -0.5\right) \cdot ky \]

   if 9.9999999999999992e-72 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 97.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-71}:\\ \;\;\;\;\left(\frac{\left(ky \cdot ky\right) \cdot \sin th}{\left(kx \cdot kx\right) \cdot kx} \cdot -0.5\right) \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 5e-9)
   (*
    (/
     (sin ky)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin th))
   (*
    (/
     (sin th)
     (sqrt (fma (- 1.0 (cos (+ ky ky))) 0.5 (* (- 1.0 (cos (+ kx kx))) 0.5))))
    (sin ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 5e-9) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = (sin(th) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, ((1.0 - cos((kx + kx))) * 0.5)))) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 5e-9)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5)))) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 5e-9], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.0000000000000001e-9

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.9

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if 5.0000000000000001e-9 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}} \cdot \sin ky} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 5e-9)
   (*
    (/
     (sin ky)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin th))
   (*
    (/
     (sin ky)
     (sqrt (fma (- 1.0 (cos (+ ky ky))) 0.5 (* (- 1.0 (cos (+ kx kx))) 0.5))))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 5e-9) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt(fma((1.0 - cos((ky + ky))), 0.5, ((1.0 - cos((kx + kx))) * 0.5)))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 5e-9)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(fma(Float64(1.0 - cos(Float64(ky + ky))), 0.5, Float64(Float64(1.0 - cos(Float64(kx + kx))) * 0.5)))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 5e-9], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.0000000000000001e-9

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.9

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if 5.0000000000000001e-9 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      4. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      8. div-inv N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      11. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(2 \cdot ky\right), \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. count-2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(ky + ky\right)}, \frac{1}{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      17. lower-+.f64 99.4

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(ky + ky\right)}, 0.5, {\sin kx}^{2}\right)}} \cdot \sin th \]

      18. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      19. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      20. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      21. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      22. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), \frac{1}{2}, \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}\right)}} \cdot \sin th \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 15.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sin th \leq 8 \cdot 10^{-297}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;1 \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))
       (sin th))
      8e-297)
   (* (* (* th th) th) -0.16666666666666666)
   (* 1.0 th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) * sin(th)) <= 8e-297) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = 1.0 * th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) * sin(th)) <= 8d-297) then
        tmp = ((th * th) * th) * (-0.16666666666666666d0)
    else
        tmp = 1.0d0 * th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) * Math.sin(th)) <= 8e-297) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = 1.0 * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) * math.sin(th)) <= 8e-297:
		tmp = ((th * th) * th) * -0.16666666666666666
	else:
		tmp = 1.0 * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) * sin(th)) <= 8e-297)
		tmp = Float64(Float64(Float64(th * th) * th) * -0.16666666666666666);
	else
		tmp = Float64(1.0 * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) * sin(th)) <= 8e-297)
		tmp = ((th * th) * th) * -0.16666666666666666;
	else
		tmp = 1.0 * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 8e-297], N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[(1.0 * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 8.00000000000000032e-297

   1. Initial program 96.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 21.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.3%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 11.7%

       \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   if 8.00000000000000032e-297 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 22.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 22.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.5%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around 0

      \[\leadsto 1 \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 15.6%

       \[\leadsto 1 \cdot th \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sin th \leq 8 \cdot 10^{-297}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;1 \cdot th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 30.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-61}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 1e-61)
   (* (* (* th th) th) -0.16666666666666666)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 1e-61) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 1d-61) then
        tmp = ((th * th) * th) * (-0.16666666666666666d0)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 1e-61) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 1e-61:
		tmp = ((th * th) * th) * -0.16666666666666666
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-61)
		tmp = Float64(Float64(Float64(th * th) * th) * -0.16666666666666666);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 1e-61)
		tmp = ((th * th) * th) * -0.16666666666666666;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-61], N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1e-61

   1. Initial program 95.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.1%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 10.5%

       \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   if 1e-61 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 97.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-61}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 67.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 170000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 170000000000.0)
   (*
    (/
     (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
     (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
    (sin th))
   (/ (* (sin th) (sin ky)) (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 170000000000.0) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else {
		tmp = (sin(th) * sin(ky)) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 170000000000.0)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) * sin(ky)) / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 170000000000.0], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.7e11

   1. Initial program 95.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 61.5

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 61.5%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 62.1

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   10. Applied rewrites 62.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   if 1.7e11 < ky

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 99.7

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky + ky\right), 0.5, \left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5\right)}}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}}{\sin ky}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}}{\sin ky}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}}{\sin ky}} \]

      10. lower-*.f64 65.0

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}}{\sin ky}} \]

   7. Applied rewrites 65.0%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}}{\sin ky}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}{\sin ky}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \cdot \sin ky} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot \frac{1}{2}}} \]

      8. lower-*.f64 65.1

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}} \]

   9. Applied rewrites 11.1%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(kx + kx\right)\right) \cdot 0.5}}} \]

   10. Taylor expanded in kx around 0

       \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{1}{2}}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot ky\right)\right) \cdot \frac{1}{2}}} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)\right)} \cdot \frac{1}{2}}} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot ky\right)\right)}\right) \cdot \frac{1}{2}}} \]

       7. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot ky\right)}\right) \cdot \frac{1}{2}}} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(\color{blue}{2} \cdot ky\right)\right) \cdot \frac{1}{2}}} \]

       9. *-commutative N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot \frac{1}{2}}} \]

       10. lower-*.f64 65.1

           \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \color{blue}{\left(ky \cdot 2\right)}\right) \cdot 0.5}} \]

   12. Applied rewrites 65.1%

       \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(1 - \cos \left(ky \cdot 2\right)\right) \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 170000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 13.6% accurate, 105.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* 1.0 th))

C

double code(double kx, double ky, double th) {
	return 1.0 * th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = 1.0d0 * th
end function

Java

public static double code(double kx, double ky, double th) {
	return 1.0 * th;
}

Python

def code(kx, ky, th):
	return 1.0 * th

Julia

function code(kx, ky, th)
	return Float64(1.0 * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 1.0 * th;
end

Wolfram

code[kx_, ky_, th_] := N[(1.0 * th), $MachinePrecision]

Derivation

1. Initial program 96.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 21.8

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 21.8%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 13.3%

   \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

9. Taylor expanded in th around 0

   \[\leadsto 1 \cdot th \]
10. Step-by-step derivation

11. Applied rewrites 13.7%

    \[\leadsto 1 \cdot th \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))