Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.9% → 99.7%

Time: 12.9s

Alternatives: 25

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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-*.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 92.6

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

   8. lift-sqrt.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 67.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(2 \cdot kx\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := 1 - \cos \left(2 \cdot ky\right)\\ t_4 := \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(t\_1 - t\_3\right)}}\\ \mathbf{if}\;t\_2 \leq -0.975:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{t\_3 \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.25:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 10^{-106}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - t\_1\right) \cdot 2} \cdot 0.5} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* 2.0 kx)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (- 1.0 (cos (* 2.0 ky))))
        (t_4 (* (* 2.0 (* (sin ky) th)) (sqrt (/ 0.5 (- 1.0 (- t_1 t_3)))))))
   (if (<= t_2 -0.975)
     (* (/ (sin ky) (/ (sqrt (* t_3 2.0)) 2.0)) (sin th))
     (if (<= t_2 -0.25)
       t_4
       (if (<= t_2 1e-106)
         (* (/ (sin th) (* (sqrt (* (- 1.0 t_1) 2.0)) 0.5)) (sin ky))
         (if (<= t_2 0.002)
           (* (/ ky (sin kx)) (sin th))
           (if (<= t_2 0.995) t_4 (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((2.0 * kx));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = 1.0 - cos((2.0 * ky));
	double t_4 = (2.0 * (sin(ky) * th)) * sqrt((0.5 / (1.0 - (t_1 - t_3))));
	double tmp;
	if (t_2 <= -0.975) {
		tmp = (sin(ky) / (sqrt((t_3 * 2.0)) / 2.0)) * sin(th);
	} else if (t_2 <= -0.25) {
		tmp = t_4;
	} else if (t_2 <= 1e-106) {
		tmp = (sin(th) / (sqrt(((1.0 - t_1) * 2.0)) * 0.5)) * sin(ky);
	} else if (t_2 <= 0.002) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_4;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = cos((2.0d0 * kx))
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    t_3 = 1.0d0 - cos((2.0d0 * ky))
    t_4 = (2.0d0 * (sin(ky) * th)) * sqrt((0.5d0 / (1.0d0 - (t_1 - t_3))))
    if (t_2 <= (-0.975d0)) then
        tmp = (sin(ky) / (sqrt((t_3 * 2.0d0)) / 2.0d0)) * sin(th)
    else if (t_2 <= (-0.25d0)) then
        tmp = t_4
    else if (t_2 <= 1d-106) then
        tmp = (sin(th) / (sqrt(((1.0d0 - t_1) * 2.0d0)) * 0.5d0)) * sin(ky)
    else if (t_2 <= 0.002d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else if (t_2 <= 0.995d0) then
        tmp = t_4
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.cos((2.0 * kx));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_3 = 1.0 - Math.cos((2.0 * ky));
	double t_4 = (2.0 * (Math.sin(ky) * th)) * Math.sqrt((0.5 / (1.0 - (t_1 - t_3))));
	double tmp;
	if (t_2 <= -0.975) {
		tmp = (Math.sin(ky) / (Math.sqrt((t_3 * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_2 <= -0.25) {
		tmp = t_4;
	} else if (t_2 <= 1e-106) {
		tmp = (Math.sin(th) / (Math.sqrt(((1.0 - t_1) * 2.0)) * 0.5)) * Math.sin(ky);
	} else if (t_2 <= 0.002) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_4;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.cos((2.0 * kx))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_3 = 1.0 - math.cos((2.0 * ky))
	t_4 = (2.0 * (math.sin(ky) * th)) * math.sqrt((0.5 / (1.0 - (t_1 - t_3))))
	tmp = 0
	if t_2 <= -0.975:
		tmp = (math.sin(ky) / (math.sqrt((t_3 * 2.0)) / 2.0)) * math.sin(th)
	elif t_2 <= -0.25:
		tmp = t_4
	elif t_2 <= 1e-106:
		tmp = (math.sin(th) / (math.sqrt(((1.0 - t_1) * 2.0)) * 0.5)) * math.sin(ky)
	elif t_2 <= 0.002:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	elif t_2 <= 0.995:
		tmp = t_4
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(2.0 * kx))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(1.0 - cos(Float64(2.0 * ky)))
	t_4 = Float64(Float64(2.0 * Float64(sin(ky) * th)) * sqrt(Float64(0.5 / Float64(1.0 - Float64(t_1 - t_3)))))
	tmp = 0.0
	if (t_2 <= -0.975)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(t_3 * 2.0)) / 2.0)) * sin(th));
	elseif (t_2 <= -0.25)
		tmp = t_4;
	elseif (t_2 <= 1e-106)
		tmp = Float64(Float64(sin(th) / Float64(sqrt(Float64(Float64(1.0 - t_1) * 2.0)) * 0.5)) * sin(ky));
	elseif (t_2 <= 0.002)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_2 <= 0.995)
		tmp = t_4;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = cos((2.0 * kx));
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_3 = 1.0 - cos((2.0 * ky));
	t_4 = (2.0 * (sin(ky) * th)) * sqrt((0.5 / (1.0 - (t_1 - t_3))));
	tmp = 0.0;
	if (t_2 <= -0.975)
		tmp = (sin(ky) / (sqrt((t_3 * 2.0)) / 2.0)) * sin(th);
	elseif (t_2 <= -0.25)
		tmp = t_4;
	elseif (t_2 <= 1e-106)
		tmp = (sin(th) / (sqrt(((1.0 - t_1) * 2.0)) * 0.5)) * sin(ky);
	elseif (t_2 <= 0.002)
		tmp = (ky / sin(kx)) * sin(th);
	elseif (t_2 <= 0.995)
		tmp = t_4;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(t$95$1 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.975], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(t$95$3 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.25], t$95$4, If[LessEqual[t$95$2, 1e-106], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.002], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], t$95$4, N[Sin[th], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.974999999999999978

   1. Initial program 83.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -0.974999999999999978 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.25 or 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   7. Applied rewrites 59.2%

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

   if -0.25 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999941e-107

   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. 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

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 73.1%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 72.2

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

   9. Applied rewrites 72.2%

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

   if 9.99999999999999941e-107 < (/.f64 (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-3

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 68.9

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

   5. Applied rewrites 68.9%

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

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

   1. Initial program 84.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.6

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

   5. Applied rewrites 92.6%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.975:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.25:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-106}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} \cdot 0.5} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := {\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ t_3 := \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ t_4 := \sin \left(\frac{2 \cdot kx}{2}\right)\\ \mathbf{if}\;t\_1 \leq -0.975:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{-1}{-2 \cdot \left(t\_4 \cdot t\_4\right)}}\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (pow (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)) -1.0))
        (t_3
         (*
          (/
           (sin th)
           (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
          (sin ky)))
        (t_4 (sin (/ (* 2.0 kx) 2.0))))
   (if (<= t_1 -0.975)
     t_3
     (if (<= t_1 -0.2)
       t_2
       (if (<= t_1 4e-7)
         (*
          (* (* 2.0 (* (sqrt 0.5) ky)) (sqrt (/ -1.0 (* -2.0 (* t_4 t_4)))))
          (sin th))
         (if (<= t_1 0.995) t_2 t_3))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = pow((hypot(sin(kx), sin(ky)) / (sin(ky) * th)), -1.0);
	double t_3 = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	double t_4 = sin(((2.0 * kx) / 2.0));
	double tmp;
	if (t_1 <= -0.975) {
		tmp = t_3;
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 4e-7) {
		tmp = ((2.0 * (sqrt(0.5) * ky)) * sqrt((-1.0 / (-2.0 * (t_4 * t_4))))) * sin(th);
	} else if (t_1 <= 0.995) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)) ^ -1.0
	t_3 = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky))
	t_4 = sin(Float64(Float64(2.0 * kx) / 2.0))
	tmp = 0.0
	if (t_1 <= -0.975)
		tmp = t_3;
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 4e-7)
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(0.5) * ky)) * sqrt(Float64(-1.0 / Float64(-2.0 * Float64(t_4 * t_4))))) * sin(th));
	elseif (t_1 <= 0.995)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[th], $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[ky], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(2.0 * kx), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.975], t$95$3, If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 4e-7], N[(N[(N[(2.0 * N[(N[Sqrt[0.5], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(-2.0 * N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], t$95$2, t$95$3]]]]]]]]

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))))) < -0.974999999999999978 or 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.1%

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

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

      7. lower-/.f64 83.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if -0.974999999999999978 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.2%

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

      1. lift-*.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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 60.0

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

   7. Applied rewrites 60.0%

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

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

   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-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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in ky around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

   9. Applied rewrites 97.2%

      \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{-2 \cdot \left(\sin \left(\frac{0 - 2 \cdot kx}{2}\right) \cdot \sin \left(\frac{0 + 2 \cdot kx}{2}\right)\right)}}\right) \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.975:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{-1}{-2 \cdot \left(\sin \left(\frac{2 \cdot kx}{2}\right) \cdot \sin \left(\frac{2 \cdot kx}{2}\right)\right)}}\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \left(\frac{2 \cdot kx}{2}\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := {\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{if}\;t\_2 \leq -0.975:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{-1}{-2 \cdot \left(t\_1 \cdot t\_1\right)}}\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.9999999998:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (/ (* 2.0 kx) 2.0)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (pow (/ (hypot (sin kx) (sin ky)) (* (sin ky) th)) -1.0)))
   (if (<= t_2 -0.975)
     (* (/ (sin ky) (/ (sqrt (* (- 1.0 (cos (* 2.0 ky))) 2.0)) 2.0)) (sin th))
     (if (<= t_2 -0.2)
       t_3
       (if (<= t_2 4e-7)
         (*
          (* (* 2.0 (* (sqrt 0.5) ky)) (sqrt (/ -1.0 (* -2.0 (* t_1 t_1)))))
          (sin th))
         (if (<= t_2 0.9999999998) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(((2.0 * kx) / 2.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = pow((hypot(sin(kx), sin(ky)) / (sin(ky) * th)), -1.0);
	double tmp;
	if (t_2 <= -0.975) {
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 4e-7) {
		tmp = ((2.0 * (sqrt(0.5) * ky)) * sqrt((-1.0 / (-2.0 * (t_1 * t_1))))) * sin(th);
	} else if (t_2 <= 0.9999999998) {
		tmp = t_3;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(((2.0 * kx) / 2.0));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_3 = Math.pow((Math.hypot(Math.sin(kx), Math.sin(ky)) / (Math.sin(ky) * th)), -1.0);
	double tmp;
	if (t_2 <= -0.975) {
		tmp = (Math.sin(ky) / (Math.sqrt(((1.0 - Math.cos((2.0 * ky))) * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 4e-7) {
		tmp = ((2.0 * (Math.sqrt(0.5) * ky)) * Math.sqrt((-1.0 / (-2.0 * (t_1 * t_1))))) * Math.sin(th);
	} else if (t_2 <= 0.9999999998) {
		tmp = t_3;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(((2.0 * kx) / 2.0))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_3 = math.pow((math.hypot(math.sin(kx), math.sin(ky)) / (math.sin(ky) * th)), -1.0)
	tmp = 0
	if t_2 <= -0.975:
		tmp = (math.sin(ky) / (math.sqrt(((1.0 - math.cos((2.0 * ky))) * 2.0)) / 2.0)) * math.sin(th)
	elif t_2 <= -0.2:
		tmp = t_3
	elif t_2 <= 4e-7:
		tmp = ((2.0 * (math.sqrt(0.5) * ky)) * math.sqrt((-1.0 / (-2.0 * (t_1 * t_1))))) * math.sin(th)
	elif t_2 <= 0.9999999998:
		tmp = t_3
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(Float64(Float64(2.0 * kx) / 2.0))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(hypot(sin(kx), sin(ky)) / Float64(sin(ky) * th)) ^ -1.0
	tmp = 0.0
	if (t_2 <= -0.975)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 2.0)) / 2.0)) * sin(th));
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 4e-7)
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(0.5) * ky)) * sqrt(Float64(-1.0 / Float64(-2.0 * Float64(t_1 * t_1))))) * sin(th));
	elseif (t_2 <= 0.9999999998)
		tmp = t_3;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(((2.0 * kx) / 2.0));
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_3 = (hypot(sin(kx), sin(ky)) / (sin(ky) * th)) ^ -1.0;
	tmp = 0.0;
	if (t_2 <= -0.975)
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 4e-7)
		tmp = ((2.0 * (sqrt(0.5) * ky)) * sqrt((-1.0 / (-2.0 * (t_1 * t_1))))) * sin(th);
	elseif (t_2 <= 0.9999999998)
		tmp = t_3;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[(N[(2.0 * kx), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[t$95$2, -0.975], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], t$95$3, If[LessEqual[t$95$2, 4e-7], N[(N[(N[(2.0 * N[(N[Sqrt[0.5], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(-2.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999999998], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.974999999999999978

   1. Initial program 83.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -0.974999999999999978 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 3.9999999999999998e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.9999999998

   1. Initial program 99.2%

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

      1. lift-*.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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 60.8

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

   7. Applied rewrites 60.8%

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

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

   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-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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in ky around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

   9. Applied rewrites 97.2%

      \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{-2 \cdot \left(\sin \left(\frac{0 - 2 \cdot kx}{2}\right) \cdot \sin \left(\frac{0 + 2 \cdot kx}{2}\right)\right)}}\right) \cdot \sin th \]

   if 0.9999999998 < (/.f64 (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 84.5%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 93.1

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

   5. Applied rewrites 93.1%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.975:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{-1}{-2 \cdot \left(\sin \left(\frac{2 \cdot kx}{2}\right) \cdot \sin \left(\frac{2 \cdot kx}{2}\right)\right)}}\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9999999998:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot th\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.975:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.25:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{t\_1}\right)}^{-1}\\ \mathbf{elif}\;t\_3 \leq 0.002:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\left(2 \cdot t\_1\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) th))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.975)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_2))) (sin th))
     (if (<= t_3 -0.25)
       (pow (/ (hypot (sin kx) (sin ky)) t_1) -1.0)
       (if (<= t_3 0.002)
         (/
          (sin th)
          (/
           (hypot (sin ky) (sin kx))
           (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
         (if (<= t_3 0.995)
           (*
            (* 2.0 t_1)
            (sqrt
             (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) (- 1.0 (cos (* 2.0 ky))))))))
           (*
            (/
             (sin th)
             (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
            (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * th;
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.975) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	} else if (t_3 <= -0.25) {
		tmp = pow((hypot(sin(kx), sin(ky)) / t_1), -1.0);
	} else if (t_3 <= 0.002) {
		tmp = sin(th) / (hypot(sin(ky), sin(kx)) / (fma((ky * ky), -0.16666666666666666, 1.0) * ky));
	} else if (t_3 <= 0.995) {
		tmp = (2.0 * t_1) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	} else {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * th)
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.975)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_2))) * sin(th));
	elseif (t_3 <= -0.25)
		tmp = Float64(hypot(sin(kx), sin(ky)) / t_1) ^ -1.0;
	elseif (t_3 <= 0.002)
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), sin(kx)) / Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)));
	elseif (t_3 <= 0.995)
		tmp = Float64(Float64(2.0 * t_1) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - Float64(1.0 - cos(Float64(2.0 * ky))))))));
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.975], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.25], N[Power[N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / t$95$1), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$3, 0.002], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], N[(N[(2.0 * t$95$1), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $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[ky], $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))))) < -0.974999999999999978

   1. Initial program 83.4%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -0.974999999999999978 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.25

   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. 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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 56.6

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

   7. Applied rewrites 56.6%

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

   if -0.25 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-3

   1. Initial program 99.4%

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

      1. lift-*.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

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   7. Applied rewrites 60.8%

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

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

   1. Initial program 84.8%

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

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

      7. lower-/.f64 84.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.975:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.25:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot th\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.975:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.25:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{t\_1}\right)}^{-1}\\ \mathbf{elif}\;t\_3 \leq 0.002:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\left(2 \cdot t\_1\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) th))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.975)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_2))) (sin th))
     (if (<= t_3 -0.25)
       (pow (/ (hypot (sin kx) (sin ky)) t_1) -1.0)
       (if (<= t_3 0.002)
         (*
          (/ (sin th) (hypot (sin ky) (sin kx)))
          (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
         (if (<= t_3 0.995)
           (*
            (* 2.0 t_1)
            (sqrt
             (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) (- 1.0 (cos (* 2.0 ky))))))))
           (*
            (/
             (sin th)
             (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
            (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * th;
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.975) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	} else if (t_3 <= -0.25) {
		tmp = pow((hypot(sin(kx), sin(ky)) / t_1), -1.0);
	} else if (t_3 <= 0.002) {
		tmp = (sin(th) / hypot(sin(ky), sin(kx))) * (fma((ky * ky), -0.16666666666666666, 1.0) * ky);
	} else if (t_3 <= 0.995) {
		tmp = (2.0 * t_1) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	} else {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * th)
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.975)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_2))) * sin(th));
	elseif (t_3 <= -0.25)
		tmp = Float64(hypot(sin(kx), sin(ky)) / t_1) ^ -1.0;
	elseif (t_3 <= 0.002)
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), sin(kx))) * Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky));
	elseif (t_3 <= 0.995)
		tmp = Float64(Float64(2.0 * t_1) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - Float64(1.0 - cos(Float64(2.0 * ky))))))));
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.975], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.25], N[Power[N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / t$95$1), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$3, 0.002], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], N[(N[(2.0 * t$95$1), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $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[ky], $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))))) < -0.974999999999999978

   1. Initial program 83.4%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -0.974999999999999978 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.25

   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. 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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 56.6

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

   7. Applied rewrites 56.6%

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

   if -0.25 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-3

   1. Initial program 99.4%

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

      1. lift-*.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

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   7. Applied rewrites 60.8%

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

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

   1. Initial program 84.8%

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

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

      7. lower-/.f64 84.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.975:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.25:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ t_3 := \sin ky \cdot th\\ \mathbf{if}\;t\_1 \leq -0.975:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -0.25:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{t\_3}\right)}^{-1}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;\left(2 \cdot t\_3\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2
         (*
          (/
           (sin th)
           (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
          (sin ky)))
        (t_3 (* (sin ky) th)))
   (if (<= t_1 -0.975)
     t_2
     (if (<= t_1 -0.25)
       (pow (/ (hypot (sin kx) (sin ky)) t_3) -1.0)
       (if (<= t_1 0.002)
         (*
          (/ (sin th) (hypot (sin ky) (sin kx)))
          (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
         (if (<= t_1 0.995)
           (*
            (* 2.0 t_3)
            (sqrt
             (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) (- 1.0 (cos (* 2.0 ky))))))))
           t_2))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	double t_3 = sin(ky) * th;
	double tmp;
	if (t_1 <= -0.975) {
		tmp = t_2;
	} else if (t_1 <= -0.25) {
		tmp = pow((hypot(sin(kx), sin(ky)) / t_3), -1.0);
	} else if (t_1 <= 0.002) {
		tmp = (sin(th) / hypot(sin(ky), sin(kx))) * (fma((ky * ky), -0.16666666666666666, 1.0) * ky);
	} else if (t_1 <= 0.995) {
		tmp = (2.0 * t_3) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky))
	t_3 = Float64(sin(ky) * th)
	tmp = 0.0
	if (t_1 <= -0.975)
		tmp = t_2;
	elseif (t_1 <= -0.25)
		tmp = Float64(hypot(sin(kx), sin(ky)) / t_3) ^ -1.0;
	elseif (t_1 <= 0.002)
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), sin(kx))) * Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky));
	elseif (t_1 <= 0.995)
		tmp = Float64(Float64(2.0 * t_3) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - Float64(1.0 - cos(Float64(2.0 * ky))))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[th], $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[ky], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]}, If[LessEqual[t$95$1, -0.975], t$95$2, If[LessEqual[t$95$1, -0.25], N[Power[N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / t$95$3), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], N[(N[(2.0 * t$95$3), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

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))))) < -0.974999999999999978 or 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.1%

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

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

      7. lower-/.f64 83.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if -0.974999999999999978 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.25

   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. 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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 56.6

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

   7. Applied rewrites 56.6%

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

   if -0.25 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-3

   1. Initial program 99.4%

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

      1. lift-*.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

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   7. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.975:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.25:\\ \;\;\;\;{\left(\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \left(\frac{2 \cdot kx}{2}\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := 1 - \cos \left(2 \cdot ky\right)\\ t_4 := \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - t\_3\right)}}\\ \mathbf{if}\;t\_2 \leq -0.975:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{t\_3 \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0.002:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{-1}{-2 \cdot \left(t\_1 \cdot t\_1\right)}}\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (/ (* 2.0 kx) 2.0)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (- 1.0 (cos (* 2.0 ky))))
        (t_4
         (*
          (* 2.0 (* (sin ky) th))
          (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) t_3)))))))
   (if (<= t_2 -0.975)
     (* (/ (sin ky) (/ (sqrt (* t_3 2.0)) 2.0)) (sin th))
     (if (<= t_2 -0.2)
       t_4
       (if (<= t_2 0.002)
         (*
          (* (* 2.0 (* (sqrt 0.5) ky)) (sqrt (/ -1.0 (* -2.0 (* t_1 t_1)))))
          (sin th))
         (if (<= t_2 0.995) t_4 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(((2.0 * kx) / 2.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = 1.0 - cos((2.0 * ky));
	double t_4 = (2.0 * (sin(ky) * th)) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - t_3))));
	double tmp;
	if (t_2 <= -0.975) {
		tmp = (sin(ky) / (sqrt((t_3 * 2.0)) / 2.0)) * sin(th);
	} else if (t_2 <= -0.2) {
		tmp = t_4;
	} else if (t_2 <= 0.002) {
		tmp = ((2.0 * (sqrt(0.5) * ky)) * sqrt((-1.0 / (-2.0 * (t_1 * t_1))))) * sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_4;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sin(((2.0d0 * kx) / 2.0d0))
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    t_3 = 1.0d0 - cos((2.0d0 * ky))
    t_4 = (2.0d0 * (sin(ky) * th)) * sqrt((0.5d0 / (1.0d0 - (cos((2.0d0 * kx)) - t_3))))
    if (t_2 <= (-0.975d0)) then
        tmp = (sin(ky) / (sqrt((t_3 * 2.0d0)) / 2.0d0)) * sin(th)
    else if (t_2 <= (-0.2d0)) then
        tmp = t_4
    else if (t_2 <= 0.002d0) then
        tmp = ((2.0d0 * (sqrt(0.5d0) * ky)) * sqrt(((-1.0d0) / ((-2.0d0) * (t_1 * t_1))))) * sin(th)
    else if (t_2 <= 0.995d0) then
        tmp = t_4
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(((2.0 * kx) / 2.0));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_3 = 1.0 - Math.cos((2.0 * ky));
	double t_4 = (2.0 * (Math.sin(ky) * th)) * Math.sqrt((0.5 / (1.0 - (Math.cos((2.0 * kx)) - t_3))));
	double tmp;
	if (t_2 <= -0.975) {
		tmp = (Math.sin(ky) / (Math.sqrt((t_3 * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_2 <= -0.2) {
		tmp = t_4;
	} else if (t_2 <= 0.002) {
		tmp = ((2.0 * (Math.sqrt(0.5) * ky)) * Math.sqrt((-1.0 / (-2.0 * (t_1 * t_1))))) * Math.sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_4;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(((2.0 * kx) / 2.0))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_3 = 1.0 - math.cos((2.0 * ky))
	t_4 = (2.0 * (math.sin(ky) * th)) * math.sqrt((0.5 / (1.0 - (math.cos((2.0 * kx)) - t_3))))
	tmp = 0
	if t_2 <= -0.975:
		tmp = (math.sin(ky) / (math.sqrt((t_3 * 2.0)) / 2.0)) * math.sin(th)
	elif t_2 <= -0.2:
		tmp = t_4
	elif t_2 <= 0.002:
		tmp = ((2.0 * (math.sqrt(0.5) * ky)) * math.sqrt((-1.0 / (-2.0 * (t_1 * t_1))))) * math.sin(th)
	elif t_2 <= 0.995:
		tmp = t_4
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(Float64(Float64(2.0 * kx) / 2.0))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(1.0 - cos(Float64(2.0 * ky)))
	t_4 = Float64(Float64(2.0 * Float64(sin(ky) * th)) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - t_3)))))
	tmp = 0.0
	if (t_2 <= -0.975)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(t_3 * 2.0)) / 2.0)) * sin(th));
	elseif (t_2 <= -0.2)
		tmp = t_4;
	elseif (t_2 <= 0.002)
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(0.5) * ky)) * sqrt(Float64(-1.0 / Float64(-2.0 * Float64(t_1 * t_1))))) * sin(th));
	elseif (t_2 <= 0.995)
		tmp = t_4;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(((2.0 * kx) / 2.0));
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_3 = 1.0 - cos((2.0 * ky));
	t_4 = (2.0 * (sin(ky) * th)) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - t_3))));
	tmp = 0.0;
	if (t_2 <= -0.975)
		tmp = (sin(ky) / (sqrt((t_3 * 2.0)) / 2.0)) * sin(th);
	elseif (t_2 <= -0.2)
		tmp = t_4;
	elseif (t_2 <= 0.002)
		tmp = ((2.0 * (sqrt(0.5) * ky)) * sqrt((-1.0 / (-2.0 * (t_1 * t_1))))) * sin(th);
	elseif (t_2 <= 0.995)
		tmp = t_4;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[(N[(2.0 * kx), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.975], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(t$95$3 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], t$95$4, If[LessEqual[t$95$2, 0.002], N[(N[(N[(2.0 * N[(N[Sqrt[0.5], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(-2.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], t$95$4, N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.974999999999999978

   1. Initial program 83.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 57.4

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

   7. Applied rewrites 57.4%

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

   if -0.974999999999999978 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   7. Applied rewrites 59.2%

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

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

   1. Initial program 99.4%

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

      1. lift-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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in ky around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 67.1

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

   7. Applied rewrites 67.1%

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

   9. Applied rewrites 96.7%

      \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{-2 \cdot \left(\sin \left(\frac{0 - 2 \cdot kx}{2}\right) \cdot \sin \left(\frac{0 + 2 \cdot kx}{2}\right)\right)}}\right) \cdot \sin th \]

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

   1. Initial program 84.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.6

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

   5. Applied rewrites 92.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.975:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{-1}{-2 \cdot \left(\sin \left(\frac{2 \cdot kx}{2}\right) \cdot \sin \left(\frac{2 \cdot kx}{2}\right)\right)}}\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.71:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 10^{-106}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} \cdot 0.5} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.71)
     (* (/ (sin ky) (/ (sqrt (* (- 1.0 (cos (* 2.0 ky))) 2.0)) 2.0)) (sin th))
     (if (<= t_1 1e-106)
       (*
        (/ (sin th) (* (sqrt (* (- 1.0 (cos (* 2.0 kx))) 2.0)) 0.5))
        (sin ky))
       (if (<= t_1 0.002) (* (/ ky (sin kx)) (sin th)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.71) {
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	} else if (t_1 <= 1e-106) {
		tmp = (sin(th) / (sqrt(((1.0 - cos((2.0 * kx))) * 2.0)) * 0.5)) * sin(ky);
	} else if (t_1 <= 0.002) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.71d0)) then
        tmp = (sin(ky) / (sqrt(((1.0d0 - cos((2.0d0 * ky))) * 2.0d0)) / 2.0d0)) * sin(th)
    else if (t_1 <= 1d-106) then
        tmp = (sin(th) / (sqrt(((1.0d0 - cos((2.0d0 * kx))) * 2.0d0)) * 0.5d0)) * sin(ky)
    else if (t_1 <= 0.002d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.71) {
		tmp = (Math.sin(ky) / (Math.sqrt(((1.0 - Math.cos((2.0 * ky))) * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_1 <= 1e-106) {
		tmp = (Math.sin(th) / (Math.sqrt(((1.0 - Math.cos((2.0 * kx))) * 2.0)) * 0.5)) * Math.sin(ky);
	} else if (t_1 <= 0.002) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.71:
		tmp = (math.sin(ky) / (math.sqrt(((1.0 - math.cos((2.0 * ky))) * 2.0)) / 2.0)) * math.sin(th)
	elif t_1 <= 1e-106:
		tmp = (math.sin(th) / (math.sqrt(((1.0 - math.cos((2.0 * kx))) * 2.0)) * 0.5)) * math.sin(ky)
	elif t_1 <= 0.002:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.71)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 2.0)) / 2.0)) * sin(th));
	elseif (t_1 <= 1e-106)
		tmp = Float64(Float64(sin(th) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * 2.0)) * 0.5)) * sin(ky));
	elseif (t_1 <= 0.002)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.71)
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	elseif (t_1 <= 1e-106)
		tmp = (sin(th) / (sqrt(((1.0 - cos((2.0 * kx))) * 2.0)) * 0.5)) * sin(ky);
	elseif (t_1 <= 0.002)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.71], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-106], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.70999999999999996

   1. Initial program 85.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 64.6%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 53.3

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

   7. Applied rewrites 53.3%

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

   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))))) < 9.99999999999999941e-107

   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. 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

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 75.9%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 66.9

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

   9. Applied rewrites 66.9%

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

   if 9.99999999999999941e-107 < (/.f64 (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-3

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 68.9

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

   5. Applied rewrites 68.9%

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

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

   1. Initial program 89.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 66.5

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

   5. Applied rewrites 66.5%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.71:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-106}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} \cdot 0.5} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.71:\\ \;\;\;\;\frac{\sin th}{\sqrt{2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)} \cdot 0.5} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 10^{-106}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} \cdot 0.5} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.71)
     (* (/ (sin th) (* (sqrt (* 2.0 (- 1.0 (cos (* 2.0 ky))))) 0.5)) (sin ky))
     (if (<= t_1 1e-106)
       (*
        (/ (sin th) (* (sqrt (* (- 1.0 (cos (* 2.0 kx))) 2.0)) 0.5))
        (sin ky))
       (if (<= t_1 0.002) (* (/ ky (sin kx)) (sin th)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.71) {
		tmp = (sin(th) / (sqrt((2.0 * (1.0 - cos((2.0 * ky))))) * 0.5)) * sin(ky);
	} else if (t_1 <= 1e-106) {
		tmp = (sin(th) / (sqrt(((1.0 - cos((2.0 * kx))) * 2.0)) * 0.5)) * sin(ky);
	} else if (t_1 <= 0.002) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.71d0)) then
        tmp = (sin(th) / (sqrt((2.0d0 * (1.0d0 - cos((2.0d0 * ky))))) * 0.5d0)) * sin(ky)
    else if (t_1 <= 1d-106) then
        tmp = (sin(th) / (sqrt(((1.0d0 - cos((2.0d0 * kx))) * 2.0d0)) * 0.5d0)) * sin(ky)
    else if (t_1 <= 0.002d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.71) {
		tmp = (Math.sin(th) / (Math.sqrt((2.0 * (1.0 - Math.cos((2.0 * ky))))) * 0.5)) * Math.sin(ky);
	} else if (t_1 <= 1e-106) {
		tmp = (Math.sin(th) / (Math.sqrt(((1.0 - Math.cos((2.0 * kx))) * 2.0)) * 0.5)) * Math.sin(ky);
	} else if (t_1 <= 0.002) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.71:
		tmp = (math.sin(th) / (math.sqrt((2.0 * (1.0 - math.cos((2.0 * ky))))) * 0.5)) * math.sin(ky)
	elif t_1 <= 1e-106:
		tmp = (math.sin(th) / (math.sqrt(((1.0 - math.cos((2.0 * kx))) * 2.0)) * 0.5)) * math.sin(ky)
	elif t_1 <= 0.002:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.71)
		tmp = Float64(Float64(sin(th) / Float64(sqrt(Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * ky))))) * 0.5)) * sin(ky));
	elseif (t_1 <= 1e-106)
		tmp = Float64(Float64(sin(th) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * 2.0)) * 0.5)) * sin(ky));
	elseif (t_1 <= 0.002)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.71)
		tmp = (sin(th) / (sqrt((2.0 * (1.0 - cos((2.0 * ky))))) * 0.5)) * sin(ky);
	elseif (t_1 <= 1e-106)
		tmp = (sin(th) / (sqrt(((1.0 - cos((2.0 * kx))) * 2.0)) * 0.5)) * sin(ky);
	elseif (t_1 <= 0.002)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.71], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-106], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.70999999999999996

   1. Initial program 85.3%

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

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

      7. lower-/.f64 85.2

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 64.5%

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

   7. Taylor expanded in kx around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 53.2

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

   9. Applied rewrites 53.2%

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

   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))))) < 9.99999999999999941e-107

   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. 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

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 75.9%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 66.9

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

   9. Applied rewrites 66.9%

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

   if 9.99999999999999941e-107 < (/.f64 (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-3

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 68.9

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

   5. Applied rewrites 68.9%

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

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

   1. Initial program 89.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 66.5

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

   5. Applied rewrites 66.5%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.71:\\ \;\;\;\;\frac{\sin th}{\sqrt{2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)} \cdot 0.5} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-106}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} \cdot 0.5} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.995:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(kx \cdot kx, 2, 1\right) - \cos \left(2 \cdot ky\right)\right)} \cdot 0.5}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 10^{-106}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} \cdot 0.5} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.995)
     (/
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/
       (* (sqrt (* 2.0 (- (fma (* kx kx) 2.0 1.0) (cos (* 2.0 ky))))) 0.5)
       (sin ky)))
     (if (<= t_1 1e-106)
       (*
        (/ (sin th) (* (sqrt (* (- 1.0 (cos (* 2.0 kx))) 2.0)) 0.5))
        (sin ky))
       (if (<= t_1 0.002) (* (/ ky (sin kx)) (sin th)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.995) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / ((sqrt((2.0 * (fma((kx * kx), 2.0, 1.0) - cos((2.0 * ky))))) * 0.5) / sin(ky));
	} else if (t_1 <= 1e-106) {
		tmp = (sin(th) / (sqrt(((1.0 - cos((2.0 * kx))) * 2.0)) * 0.5)) * sin(ky);
	} else if (t_1 <= 0.002) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.995)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(Float64(sqrt(Float64(2.0 * Float64(fma(Float64(kx * kx), 2.0, 1.0) - cos(Float64(2.0 * ky))))) * 0.5) / sin(ky)));
	elseif (t_1 <= 1e-106)
		tmp = Float64(Float64(sin(th) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * 2.0)) * 0.5)) * sin(ky));
	elseif (t_1 <= 0.002)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.995], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[(N[Sqrt[N[(2.0 * N[(N[(N[(kx * kx), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision] - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-106], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 82.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 82.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.4

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

   7. Applied rewrites 45.4%

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

   9. Applied rewrites 24.7%

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

   10. Taylor expanded in kx around 0

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-*.f64 24.7

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

   12. Applied rewrites 24.7%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999941e-107

   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. 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

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 78.1%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 62.2

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

   9. Applied rewrites 62.2%

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

   if 9.99999999999999941e-107 < (/.f64 (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-3

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 68.9

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

   5. Applied rewrites 68.9%

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

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

   1. Initial program 89.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 66.5

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

   5. Applied rewrites 66.5%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(kx \cdot kx, 2, 1\right) - \cos \left(2 \cdot ky\right)\right)} \cdot 0.5}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-106}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} \cdot 0.5} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)} \cdot 0.5}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 10^{-106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (/
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/ (* (sqrt (* 2.0 (- 1.0 (cos (* 2.0 ky))))) 0.5) (sin ky)))
     (if (<= t_1 1e-106)
       (*
        (/
         (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
         (/ (* (sqrt (- 1.0 (cos (* 2.0 kx)))) (sqrt 2.0)) 2.0))
        (sin th))
       (if (<= t_1 0.002) (* (/ ky (sin kx)) (sin th)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / ((sqrt((2.0 * (1.0 - cos((2.0 * ky))))) * 0.5) / sin(ky));
	} else if (t_1 <= 1e-106) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / ((sqrt((1.0 - cos((2.0 * kx)))) * sqrt(2.0)) / 2.0)) * sin(th);
	} else if (t_1 <= 0.002) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(Float64(sqrt(Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * ky))))) * 0.5) / sin(ky)));
	elseif (t_1 <= 1e-106)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / Float64(Float64(sqrt(Float64(1.0 - cos(Float64(2.0 * kx)))) * sqrt(2.0)) / 2.0)) * sin(th));
	elseif (t_1 <= 0.002)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[(N[Sqrt[N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-106], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[(N[Sqrt[N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 87.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 87.2

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 46.8

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

   7. Applied rewrites 46.8%

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

   9. Applied rewrites 31.4%

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

   10. Taylor expanded in kx around 0

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

       1. lower--.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-*.f64 21.4

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

   12. Applied rewrites 21.4%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999941e-107

   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-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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 73.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      5. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \color{blue}{\cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \color{blue}{\left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      7. lower-sqrt.f64 72.1

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \color{blue}{\sqrt{2}}}{2}} \cdot \sin th \]

   7. Applied rewrites 72.1%

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      7. lower-*.f64 71.8

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

   10. Applied rewrites 71.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\frac{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

   if 9.99999999999999941e-107 < (/.f64 (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-3

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 68.9

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 53.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)} \cdot 0.5}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 10^{-106}:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{{\left(1 - \cos \left(2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (/
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/ (* (sqrt (* 2.0 (- 1.0 (cos (* 2.0 ky))))) 0.5) (sin ky)))
     (if (<= t_1 1e-106)
       (*
        (*
         (* 2.0 (* (sqrt 0.5) ky))
         (sqrt (pow (- 1.0 (cos (* 2.0 kx))) -1.0)))
        (sin th))
       (if (<= t_1 0.002) (* (/ ky (sin kx)) (sin th)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / ((sqrt((2.0 * (1.0 - cos((2.0 * ky))))) * 0.5) / sin(ky));
	} else if (t_1 <= 1e-106) {
		tmp = ((2.0 * (sqrt(0.5) * ky)) * sqrt(pow((1.0 - cos((2.0 * kx))), -1.0))) * sin(th);
	} else if (t_1 <= 0.002) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(Float64(sqrt(Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * ky))))) * 0.5) / sin(ky)));
	elseif (t_1 <= 1e-106)
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(0.5) * ky)) * sqrt((Float64(1.0 - cos(Float64(2.0 * kx))) ^ -1.0))) * sin(th));
	elseif (t_1 <= 0.002)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[(N[Sqrt[N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-106], N[(N[(N[(2.0 * N[(N[Sqrt[0.5], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 87.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 87.2

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}}{\sin ky}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}}{\sin ky}} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \]

      15. lower-hypot.f64 99.8

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th}}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th}}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

      7. lower-*.f64 46.8

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

   7. Applied rewrites 46.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]

   9. Applied rewrites 31.4%

      \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\color{blue}{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)} \cdot 0.5}}{\sin ky}} \]

   10. Taylor expanded in kx around 0

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{2 \cdot \color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right)}} \cdot \frac{1}{2}}{\sin ky}} \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{2 \cdot \color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right)}} \cdot \frac{1}{2}}{\sin ky}} \]

       2. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sqrt{2 \cdot \left(1 - \color{blue}{\cos \left(2 \cdot ky\right)}\right)} \cdot \frac{1}{2}}{\sin ky}} \]

       3. lower-*.f64 21.4

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{2 \cdot \left(1 - \cos \color{blue}{\left(2 \cdot ky\right)}\right)} \cdot 0.5}{\sin ky}} \]

   12. Applied rewrites 21.4%

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{2 \cdot \color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right)}} \cdot 0.5}{\sin ky}} \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999941e-107

   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-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 73.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(\left(ky \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right)} \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot ky\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot ky\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{\sqrt{\frac{1}{2}}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \color{blue}{\cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      11. lower-*.f64 71.7

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \color{blue}{\left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

   7. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

   if 9.99999999999999941e-107 < (/.f64 (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-3

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 68.9

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sqrt{2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)} \cdot 0.5}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-106}:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{{\left(1 - \cos \left(2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{0.08333333333333333}{\sqrt{0.5}}, kx \cdot kx, \sqrt{0.5}\right)}{kx}\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 2e-96)
     (* (* (* -0.16666666666666666 th) th) th)
     (if (<= t_1 0.002)
       (*
        (*
         (* 2.0 (* (sqrt 0.5) ky))
         (/ (fma (/ 0.08333333333333333 (sqrt 0.5)) (* kx kx) (sqrt 0.5)) kx))
        (* (fma (* th th) -0.16666666666666666 1.0) th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 2e-96) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else if (t_1 <= 0.002) {
		tmp = ((2.0 * (sqrt(0.5) * ky)) * (fma((0.08333333333333333 / sqrt(0.5)), (kx * kx), sqrt(0.5)) / kx)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 2e-96)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	elseif (t_1 <= 0.002)
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(0.5) * ky)) * Float64(fma(Float64(0.08333333333333333 / sqrt(0.5)), Float64(kx * kx), sqrt(0.5)) / kx)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * 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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-96], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(N[(N[(2.0 * N[(N[Sqrt[0.5], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.08333333333333333 / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.9999999999999998e-96

   1. Initial program 93.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.6%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.3%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 14.3%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 14.3%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 1.9999999999999998e-96 < (/.f64 (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-3

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 38.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(\left(ky \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right)} \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot ky\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot ky\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{\sqrt{\frac{1}{2}}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \color{blue}{\cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      11. lower-*.f64 37.2

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \color{blue}{\left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

   7. Applied rewrites 37.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \frac{\sqrt{\frac{1}{2}} + \frac{1}{12} \cdot \frac{{kx}^{2}}{\sqrt{\frac{1}{2}}}}{\color{blue}{kx}}\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 39.6%

       \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{0.08333333333333333}{\sqrt{0.5}}, kx \cdot kx, \sqrt{0.5}\right)}{\color{blue}{kx}}\right) \cdot \sin th \]

   11. Taylor expanded in th around 0

       \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, kx \cdot kx, \sqrt{\frac{1}{2}}\right)}{kx}\right) \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, kx \cdot kx, \sqrt{\frac{1}{2}}\right)}{kx}\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, kx \cdot kx, \sqrt{\frac{1}{2}}\right)}{kx}\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

       3. +-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, kx \cdot kx, \sqrt{\frac{1}{2}}\right)}{kx}\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, kx \cdot kx, \sqrt{\frac{1}{2}}\right)}{kx}\right) \cdot \left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, kx \cdot kx, \sqrt{\frac{1}{2}}\right)}{kx}\right) \cdot \left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th\right) \]

       6. unpow2 N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{\frac{1}{12}}{\sqrt{\frac{1}{2}}}, kx \cdot kx, \sqrt{\frac{1}{2}}\right)}{kx}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th\right) \]

       7. lower-*.f64 25.3

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{0.08333333333333333}{\sqrt{0.5}}, kx \cdot kx, \sqrt{0.5}\right)}{kx}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th\right) \]

   13. Applied rewrites 25.3%

       \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{0.08333333333333333}{\sqrt{0.5}}, kx \cdot kx, \sqrt{0.5}\right)}{kx}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]

   if 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 38.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08888888888888889, kx \cdot kx, -0.6666666666666666\right), kx \cdot kx, 2\right) \cdot \left(kx \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.002)
   (*
    (*
     (* 2.0 (* (sqrt 0.5) ky))
     (sqrt
      (pow
       (*
        (fma
         (fma 0.08888888888888889 (* kx kx) -0.6666666666666666)
         (* kx kx)
         2.0)
        (* kx kx))
       -1.0)))
    (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.002) {
		tmp = ((2.0 * (sqrt(0.5) * ky)) * sqrt(pow((fma(fma(0.08888888888888889, (kx * kx), -0.6666666666666666), (kx * kx), 2.0) * (kx * kx)), -1.0))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.002)
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(0.5) * ky)) * sqrt((Float64(fma(fma(0.08888888888888889, Float64(kx * kx), -0.6666666666666666), Float64(kx * kx), 2.0) * Float64(kx * kx)) ^ -1.0))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(2.0 * N[(N[Sqrt[0.5], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(N[(0.08888888888888889 * N[(kx * kx), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 2.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-3

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 68.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(\left(ky \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right)} \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot ky\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot ky\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{\sqrt{\frac{1}{2}}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \color{blue}{\cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      11. lower-*.f64 38.3

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \color{blue}{\left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

   7. Applied rewrites 38.3%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{{kx}^{2} \cdot \left(2 + {kx}^{2} \cdot \left(\frac{4}{45} \cdot {kx}^{2} - \frac{2}{3}\right)\right)}}\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 29.9%

       \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.08888888888888889, kx \cdot kx, -0.6666666666666666\right), kx \cdot kx, 2\right) \cdot \left(kx \cdot kx\right)}}\right) \cdot \sin th \]

   if 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08888888888888889, kx \cdot kx, -0.6666666666666666\right), kx \cdot kx, 2\right) \cdot \left(kx \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.002)
   (/ (sin th) (/ (sin kx) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.002) {
		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(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.002d0) 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(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.002) {
		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(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.002:
		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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.002)
		tmp = Float64(sin(th) / Float64(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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.002)
		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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $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))))) < 2e-3

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      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 93.9

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}}{\sin ky}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}}{\sin ky}} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \]

      15. lower-hypot.f64 99.7

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

      2. lower-sin.f64 38.5

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sin kx}}{ky}} \]

   7. Applied rewrites 38.5%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

   if 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 43.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.002)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.002) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.002d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.002) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.002:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.002)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.002)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-3

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 38.5

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 35.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\left(\frac{0.5 \cdot ky}{kx} \cdot 2\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.002)
   (* (* (/ (* 0.5 ky) kx) 2.0) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.002) {
		tmp = (((0.5 * ky) / kx) * 2.0) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.002d0) then
        tmp = (((0.5d0 * ky) / kx) * 2.0d0) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.002) {
		tmp = (((0.5 * ky) / kx) * 2.0) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.002:
		tmp = (((0.5 * ky) / kx) * 2.0) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.002)
		tmp = Float64(Float64(Float64(Float64(0.5 * ky) / kx) * 2.0) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.002)
		tmp = (((0.5 * ky) / kx) * 2.0) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(N[(0.5 * ky), $MachinePrecision] / kx), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-3

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 68.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(\left(ky \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right)} \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot ky\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot ky\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{\sqrt{\frac{1}{2}}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \color{blue}{\cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      11. lower-*.f64 38.3

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \color{blue}{\left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

   7. Applied rewrites 38.3%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(2 \cdot \color{blue}{\frac{ky \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2}}{kx}}\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 26.4%

       \[\leadsto \left(\frac{0.5 \cdot ky}{kx} \cdot \color{blue}{2}\right) \cdot \sin th \]

   if 2e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 15.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 5 \cdot 10^{-313}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;1 \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      5e-313)
   (* (* (* -0.16666666666666666 th) th) th)
   (* 1.0 th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 5e-313) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} 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(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)) <= 5d-313) then
        tmp = (((-0.16666666666666666d0) * th) * th) * th
    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(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th)) <= 5e-313) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} 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(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)) <= 5e-313:
		tmp = ((-0.16666666666666666 * th) * th) * th
	else:
		tmp = 1.0 * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 5e-313)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 5e-313)
		tmp = ((-0.16666666666666666 * th) * th) * th;
	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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 5e-313], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $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)) < 5.00000000002e-313

   1. Initial program 93.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 23.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 23.8%

      \[\leadsto \color{blue}{\sin th} \]

   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 15.2%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 17.1%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 17.1%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 5.00000000002e-313 < (*.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 92.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 24.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 24.6%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.8%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 3.7%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   12. Taylor expanded in th around 0

       \[\leadsto 1 \cdot th \]
   13. Step-by-step derivation

   14. Applied rewrites 15.4%

       \[\leadsto 1 \cdot th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin th) (hypot (sin ky) (sin kx))) (sin ky)))

C

double code(double kx, double ky, double th) {
	return (sin(th) / hypot(sin(ky), sin(kx))) * sin(ky);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(th) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(ky);
}

Python

def code(kx, ky, th):
	return (math.sin(th) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(ky)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(th) / hypot(sin(ky), sin(kx))) * sin(ky))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(th) / hypot(sin(ky), sin(kx))) * sin(ky);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]

Derivation

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-*.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

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

   7. lower-/.f64 92.6

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

   10. +-commutative N/A

       \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

   11. lift-pow.f64 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

   12. unpow2 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

   14. unpow2 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

   15. lower-hypot.f64 99.6

       \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
5. Add Preprocessing

Alternative 21: 74.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.00155:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \cos \left(kx + kx\right)\right) + 1\right) - \cos \left(ky + ky\right)\right)} \cdot 0.5} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.00155)
   (/
    (sin th)
    (/
     (hypot (sin ky) (sin kx))
     (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (*
    (/
     (sin th)
     (*
      (sqrt (* 2.0 (- (+ (- 1.0 (cos (+ kx kx))) 1.0) (cos (+ ky ky)))))
      0.5))
    (sin ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.00155) {
		tmp = sin(th) / (hypot(sin(ky), sin(kx)) / (fma((ky * ky), -0.16666666666666666, 1.0) * ky));
	} else {
		tmp = (sin(th) / (sqrt((2.0 * (((1.0 - cos((kx + kx))) + 1.0) - cos((ky + ky))))) * 0.5)) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.00155)
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), sin(kx)) / Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)));
	else
		tmp = Float64(Float64(sin(th) / Float64(sqrt(Float64(2.0 * Float64(Float64(Float64(1.0 - cos(Float64(kx + kx))) + 1.0) - cos(Float64(ky + ky))))) * 0.5)) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.00155], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(2.0 * N[(N[(N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.00154999999999999995

   1. Initial program 90.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-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 90.6

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}}{\sin ky}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}}{\sin ky}} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \]

      15. lower-hypot.f64 99.7

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}} \]

      7. lower-*.f64 71.2

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}} \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}} \]

   if 0.00154999999999999995 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      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

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.6

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   6. Applied rewrites 99.0%

      \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)} \cdot 0.5}} \cdot \sin ky \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \color{blue}{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}} \cdot \frac{1}{2}} \cdot \sin ky \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right)}\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      3. associate-+r- N/A

         \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \color{blue}{\left(\left(\left(1 - \cos \left(2 \cdot kx\right)\right) + 1\right) - \cos \left(2 \cdot ky\right)\right)}} \cdot \frac{1}{2}} \cdot \sin ky \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \color{blue}{\left(\left(\left(1 - \cos \left(2 \cdot kx\right)\right) + 1\right) - \cos \left(2 \cdot ky\right)\right)}} \cdot \frac{1}{2}} \cdot \sin ky \]

      5. lower-+.f64 99.0

         \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\color{blue}{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) + 1\right)} - \cos \left(2 \cdot ky\right)\right)} \cdot 0.5} \cdot \sin ky \]

      6. lift-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}\right) + 1\right) - \cos \left(2 \cdot ky\right)\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}\right) + 1\right) - \cos \left(2 \cdot ky\right)\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      8. cos-2 N/A

         \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \color{blue}{\left(\cos kx \cdot \cos kx - \sin kx \cdot \sin kx\right)}\right) + 1\right) - \cos \left(2 \cdot ky\right)\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      9. cos-sum N/A

         \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \color{blue}{\cos \left(kx + kx\right)}\right) + 1\right) - \cos \left(2 \cdot ky\right)\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \color{blue}{\cos \left(kx + kx\right)}\right) + 1\right) - \cos \left(2 \cdot ky\right)\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      11. lower-+.f64 99.0

          \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \cos \color{blue}{\left(kx + kx\right)}\right) + 1\right) - \cos \left(2 \cdot ky\right)\right)} \cdot 0.5} \cdot \sin ky \]

      12. lift-cos.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \cos \left(kx + kx\right)\right) + 1\right) - \color{blue}{\cos \left(2 \cdot ky\right)}\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \cos \left(kx + kx\right)\right) + 1\right) - \cos \color{blue}{\left(2 \cdot ky\right)}\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      14. cos-2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \cos \left(kx + kx\right)\right) + 1\right) - \color{blue}{\left(\cos ky \cdot \cos ky - \sin ky \cdot \sin ky\right)}\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      15. cos-sum N/A

          \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \cos \left(kx + kx\right)\right) + 1\right) - \color{blue}{\cos \left(ky + ky\right)}\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      16. lower-cos.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \cos \left(kx + kx\right)\right) + 1\right) - \color{blue}{\cos \left(ky + ky\right)}\right)} \cdot \frac{1}{2}} \cdot \sin ky \]

      17. lower-+.f64 99.0

          \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \left(\left(\left(1 - \cos \left(kx + kx\right)\right) + 1\right) - \cos \color{blue}{\left(ky + ky\right)}\right)} \cdot 0.5} \cdot \sin ky \]

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\sin th}{\sqrt{2 \cdot \color{blue}{\left(\left(\left(1 - \cos \left(kx + kx\right)\right) + 1\right) - \cos \left(ky + ky\right)\right)}} \cdot 0.5} \cdot \sin ky \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 74.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.00145:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\frac{2}{\sqrt{\left(\left(1 - \cos \left(ky + ky\right)\right) + \left(1 - \cos \left(kx + kx\right)\right)\right) \cdot 2}} \cdot \sin ky\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.00145)
   (/
    (sin th)
    (/
     (hypot (sin ky) (sin kx))
     (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (*
    (sin th)
    (*
     (/ 2.0 (sqrt (* (+ (- 1.0 (cos (+ ky ky))) (- 1.0 (cos (+ kx kx)))) 2.0)))
     (sin ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.00145) {
		tmp = sin(th) / (hypot(sin(ky), sin(kx)) / (fma((ky * ky), -0.16666666666666666, 1.0) * ky));
	} else {
		tmp = sin(th) * ((2.0 / sqrt((((1.0 - cos((ky + ky))) + (1.0 - cos((kx + kx)))) * 2.0))) * sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.00145)
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), sin(kx)) / Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)));
	else
		tmp = Float64(sin(th) * Float64(Float64(2.0 / sqrt(Float64(Float64(Float64(1.0 - cos(Float64(ky + ky))) + Float64(1.0 - cos(Float64(kx + kx)))) * 2.0))) * sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.00145], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[(2.0 / N[Sqrt[N[(N[(N[(1.0 - N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.00145

   1. Initial program 90.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-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 90.6

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}}{\sin ky}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}}{\sin ky}} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \]

      15. lower-hypot.f64 99.7

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}} \]

      7. lower-*.f64 71.2

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}} \]

   7. Applied rewrites 71.2%

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}} \]

   if 0.00145 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      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

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.6

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   6. Applied rewrites 99.0%

      \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)} \cdot 0.5}} \cdot \sin ky \]

   8. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\sin th \cdot \left(\frac{2}{\sqrt{\left(\left(1 - \cos \left(ky + ky\right)\right) + \left(1 - \cos \left(kx + kx\right)\right)\right) \cdot 2}} \cdot \sin ky\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 36.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4.2 \cdot 10^{-147}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{{\left(1 - \cos \left(2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 4.2e-147)
   (sin th)
   (if (<= kx 7.2e-5)
     (* (/ (sin ky) (sqrt (+ (* kx kx) (* ky ky)))) (sin th))
     (*
      (* (* 2.0 (* (sqrt 0.5) ky)) (sqrt (pow (- 1.0 (cos (* 2.0 kx))) -1.0)))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.2e-147) {
		tmp = sin(th);
	} else if (kx <= 7.2e-5) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	} else {
		tmp = ((2.0 * (sqrt(0.5) * ky)) * sqrt(pow((1.0 - cos((2.0 * kx))), -1.0))) * 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 (kx <= 4.2d-147) then
        tmp = sin(th)
    else if (kx <= 7.2d-5) then
        tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th)
    else
        tmp = ((2.0d0 * (sqrt(0.5d0) * ky)) * sqrt(((1.0d0 - cos((2.0d0 * kx))) ** (-1.0d0)))) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.2e-147) {
		tmp = Math.sin(th);
	} else if (kx <= 7.2e-5) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + (ky * ky)))) * Math.sin(th);
	} else {
		tmp = ((2.0 * (Math.sqrt(0.5) * ky)) * Math.sqrt(Math.pow((1.0 - Math.cos((2.0 * kx))), -1.0))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 4.2e-147:
		tmp = math.sin(th)
	elif kx <= 7.2e-5:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + (ky * ky)))) * math.sin(th)
	else:
		tmp = ((2.0 * (math.sqrt(0.5) * ky)) * math.sqrt(math.pow((1.0 - math.cos((2.0 * kx))), -1.0))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 4.2e-147)
		tmp = sin(th);
	elseif (kx <= 7.2e-5)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(ky * ky)))) * sin(th));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(0.5) * ky)) * sqrt((Float64(1.0 - cos(Float64(2.0 * kx))) ^ -1.0))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 4.2e-147)
		tmp = sin(th);
	elseif (kx <= 7.2e-5)
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	else
		tmp = ((2.0 * (sqrt(0.5) * ky)) * sqrt(((1.0 - cos((2.0 * kx))) ^ -1.0))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 4.2e-147], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 7.2e-5], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[Sqrt[0.5], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 4.2e-147

   1. Initial program 88.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 27.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\sin th} \]

   if 4.2e-147 < kx < 7.20000000000000018e-5

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

      2. lower-*.f64 62.1

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   if 7.20000000000000018e-5 < kx

   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-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(\left(ky \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(ky \cdot \sqrt{\frac{1}{2}}\right)\right)} \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot ky\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot ky\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{\sqrt{\frac{1}{2}}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{\frac{1}{2}} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \color{blue}{\cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      11. lower-*.f64 55.8

          \[\leadsto \left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \color{blue}{\left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

   7. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 4.2 \cdot 10^{-147}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \left(\sqrt{0.5} \cdot ky\right)\right) \cdot \sqrt{{\left(1 - \cos \left(2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 36.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4.2 \cdot 10^{-147}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{{\left(1 - \cos \left(2 \cdot kx\right)\right)}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 4.2e-147)
   (sin th)
   (if (<= kx 8.2e-5)
     (* (/ (sin ky) (sqrt (+ (* kx kx) (* ky ky)))) (sin th))
     (*
      (* 2.0 (* (* (sin th) ky) (sqrt 0.5)))
      (sqrt (pow (- 1.0 (cos (* 2.0 kx))) -1.0))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.2e-147) {
		tmp = sin(th);
	} else if (kx <= 8.2e-5) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	} else {
		tmp = (2.0 * ((sin(th) * ky) * sqrt(0.5))) * sqrt(pow((1.0 - cos((2.0 * kx))), -1.0));
	}
	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 (kx <= 4.2d-147) then
        tmp = sin(th)
    else if (kx <= 8.2d-5) then
        tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th)
    else
        tmp = (2.0d0 * ((sin(th) * ky) * sqrt(0.5d0))) * sqrt(((1.0d0 - cos((2.0d0 * kx))) ** (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.2e-147) {
		tmp = Math.sin(th);
	} else if (kx <= 8.2e-5) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + (ky * ky)))) * Math.sin(th);
	} else {
		tmp = (2.0 * ((Math.sin(th) * ky) * Math.sqrt(0.5))) * Math.sqrt(Math.pow((1.0 - Math.cos((2.0 * kx))), -1.0));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 4.2e-147:
		tmp = math.sin(th)
	elif kx <= 8.2e-5:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + (ky * ky)))) * math.sin(th)
	else:
		tmp = (2.0 * ((math.sin(th) * ky) * math.sqrt(0.5))) * math.sqrt(math.pow((1.0 - math.cos((2.0 * kx))), -1.0))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 4.2e-147)
		tmp = sin(th);
	elseif (kx <= 8.2e-5)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(ky * ky)))) * sin(th));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(sin(th) * ky) * sqrt(0.5))) * sqrt((Float64(1.0 - cos(Float64(2.0 * kx))) ^ -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 4.2e-147)
		tmp = sin(th);
	elseif (kx <= 8.2e-5)
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	else
		tmp = (2.0 * ((sin(th) * ky) * sqrt(0.5))) * sqrt(((1.0 - cos((2.0 * kx))) ^ -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 4.2e-147], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 8.2e-5], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 4.2e-147

   1. Initial program 88.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 27.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 27.2%

      \[\leadsto \color{blue}{\sin th} \]

   if 4.2e-147 < kx < 8.20000000000000009e-5

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

      2. lower-*.f64 62.1

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   if 8.20000000000000009e-5 < kx

   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-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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{2 \cdot \left(\left(ky \cdot \left(\sin th \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{\frac{1}{2}}\right)\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{\frac{1}{2}}\right)\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{\frac{1}{2}}\right)\right)\right)} \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{2}}\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{2}}\right)}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      6. *-commutative N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \left(\left(\color{blue}{\sin th} \cdot ky\right) \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \color{blue}{\sqrt{\frac{1}{2}}}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]

      12. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot \sqrt{\frac{1}{1 - \color{blue}{\cos \left(2 \cdot kx\right)}}} \]

      14. lower-*.f64 55.9

          \[\leadsto \left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \color{blue}{\left(2 \cdot kx\right)}}} \]

   7. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 4.2 \cdot 10^{-147}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{{\left(1 - \cos \left(2 \cdot kx\right)\right)}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 14.2% 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 92.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 24.2

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 24.2%

   \[\leadsto \color{blue}{\sin th} \]

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 15.0%

   \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

9. Taylor expanded in th around inf

   \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
10. Step-by-step derivation

11. Applied rewrites 10.7%

    \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

12. Taylor expanded in th around 0

    \[\leadsto 1 \cdot th \]
13. Step-by-step derivation

14. Applied rewrites 15.6%

    \[\leadsto 1 \cdot th \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024315 
(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)))