Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 4.5s

Alternatives: 16

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

3. Applied rewrites 99.7%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

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

   6. lower-/.f64 94.1

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

   7. lift-sqrt.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-hypot.f64 99.6

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

3. Applied rewrites 99.6%

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

Alternative 3: 69.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \cos \left(kx + kx\right)\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.4:\\ \;\;\;\;\frac{th \cdot \sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - t\_2 \cdot 0.5\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (cos (+ kx kx))))
   (if (<= t_1 -0.99)
     (* (/ (sin ky) (sqrt (fma (sin ky) (sin ky) (- 0.5 0.5)))) (sin th))
     (if (<= t_1 -0.4)
       (/
        (* th (sin ky))
        (sqrt (- 0.5 (- (* (cos (+ ky ky)) 0.5) (- 0.5 (* t_2 0.5))))))
       (if (<= t_1 0.7)
         (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 t_2)))) (sin th))
         (if (<= t_1 2.0) (sin th) (* (/ (sin ky) (sin kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = cos((kx + kx));
	double tmp;
	if (t_1 <= -0.99) {
		tmp = (sin(ky) / sqrt(fma(sin(ky), sin(ky), (0.5 - 0.5)))) * sin(th);
	} else if (t_1 <= -0.4) {
		tmp = (th * sin(ky)) / sqrt((0.5 - ((cos((ky + ky)) * 0.5) - (0.5 - (t_2 * 0.5)))));
	} else if (t_1 <= 0.7) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * t_2)))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = cos(Float64(kx + kx))
	tmp = 0.0
	if (t_1 <= -0.99)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(sin(ky), sin(ky), Float64(0.5 - 0.5)))) * sin(th));
	elseif (t_1 <= -0.4)
		tmp = Float64(Float64(th * sin(ky)) / sqrt(Float64(0.5 - Float64(Float64(cos(Float64(ky + ky)) * 0.5) - Float64(0.5 - Float64(t_2 * 0.5))))));
	elseif (t_1 <= 0.7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * t_2)))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Sin[ky], $MachinePrecision] * N[Sin[ky], $MachinePrecision] + N[(0.5 - 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.4], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 86.2

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 65.3

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 64.9

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

   3. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-sin-a-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-fma.f64 86.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.0

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 83.8%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40000000000000002

   1. Initial program 99.3%

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 99.3

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

   3. Applied rewrites 99.3%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqr-sin-a-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sqr-sin-a-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. +-commutative N/A

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

      17. lift--.f64 N/A

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

      18. associate-+l- N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 51.0%

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

   if -0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.2

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 99.0

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 78.4

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

   3. Applied rewrites 78.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 78.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 78.4

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 78.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.4

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 78.4

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

   5. Applied rewrites 78.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 39.5

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

   7. Applied rewrites 39.5%

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

   8. Taylor expanded in ky around 0

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

      1. pow2 N/A

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

      2. sqr-sin-a-rev N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 63.8

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f64 63.8

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

   10. Applied rewrites 63.8%

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

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.6

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 80.0

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 79.5

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

   3. Applied rewrites 79.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-sin-a-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-fma.f64 99.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.2

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in kx around 0

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

      1. lift-sin.f64 78.7

         \[\leadsto \sin th \]

   8. Applied rewrites 78.7%

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

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

   1. Initial program 2.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 2.6

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 2.6

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 2.6

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

   3. Applied rewrites 2.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 2.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 2.6

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 2.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 2.6

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 2.6

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

   5. Applied rewrites 2.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 1.5

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in ky around 0

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

      1. lift-sin.f64 29.8

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

   10. Applied rewrites 29.8%

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

Alternative 4: 65.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \cos \left(kx + kx\right)\\ t_3 := \cos \left(ky + ky\right)\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot t\_3\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.4:\\ \;\;\;\;\frac{th \cdot \sin ky}{\sqrt{0.5 - \left(t\_3 \cdot 0.5 - \left(0.5 - t\_2 \cdot 0.5\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (cos (+ kx kx)))
        (t_3 (cos (+ ky ky))))
   (if (<= t_1 -0.99)
     (* (/ (sin ky) (sqrt (fma kx kx (- 0.5 (* 0.5 t_3))))) (sin th))
     (if (<= t_1 -0.4)
       (/ (* th (sin ky)) (sqrt (- 0.5 (- (* t_3 0.5) (- 0.5 (* t_2 0.5))))))
       (if (<= t_1 0.7)
         (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 t_2)))) (sin th))
         (if (<= t_1 2.0) (sin th) (* (/ (sin ky) (sin kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = cos((kx + kx));
	double t_3 = cos((ky + ky));
	double tmp;
	if (t_1 <= -0.99) {
		tmp = (sin(ky) / sqrt(fma(kx, kx, (0.5 - (0.5 * t_3))))) * sin(th);
	} else if (t_1 <= -0.4) {
		tmp = (th * sin(ky)) / sqrt((0.5 - ((t_3 * 0.5) - (0.5 - (t_2 * 0.5)))));
	} else if (t_1 <= 0.7) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * t_2)))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = cos(Float64(kx + kx))
	t_3 = cos(Float64(ky + ky))
	tmp = 0.0
	if (t_1 <= -0.99)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(kx, kx, Float64(0.5 - Float64(0.5 * t_3))))) * sin(th));
	elseif (t_1 <= -0.4)
		tmp = Float64(Float64(th * sin(ky)) / sqrt(Float64(0.5 - Float64(Float64(t_3 * 0.5) - Float64(0.5 - Float64(t_2 * 0.5))))));
	elseif (t_1 <= 0.7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * t_2)))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(kx * kx + N[(0.5 - N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.4], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[(t$95$3 * 0.5), $MachinePrecision] - N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 86.2

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 65.3

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 64.9

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

   3. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-sin-a-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-fma.f64 86.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.0

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. pow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 62.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 62.7

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

   8. Applied rewrites 62.7%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40000000000000002

   1. Initial program 99.3%

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 99.3

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

   3. Applied rewrites 99.3%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqr-sin-a-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sqr-sin-a-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. +-commutative N/A

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

      17. lift--.f64 N/A

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

      18. associate-+l- N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 51.0%

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

   if -0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.2

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 99.0

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 78.4

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

   3. Applied rewrites 78.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 78.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 78.4

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 78.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.4

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 78.4

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

   5. Applied rewrites 78.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 39.5

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

   7. Applied rewrites 39.5%

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

   8. Taylor expanded in ky around 0

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

      1. pow2 N/A

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

      2. sqr-sin-a-rev N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 63.8

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f64 63.8

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

   10. Applied rewrites 63.8%

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

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.6

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 80.0

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 79.5

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

   3. Applied rewrites 79.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-sin-a-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-fma.f64 99.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.2

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in kx around 0

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

      1. lift-sin.f64 78.7

         \[\leadsto \sin th \]

   8. Applied rewrites 78.7%

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

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

   1. Initial program 2.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 2.6

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 2.6

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 2.6

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

   3. Applied rewrites 2.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 2.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 2.6

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 2.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 2.6

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 2.6

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

   5. Applied rewrites 2.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 1.5

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in ky around 0

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

      1. lift-sin.f64 29.8

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

   10. Applied rewrites 29.8%

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

Alternative 5: 65.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \cos \left(kx + kx\right)\\ t_3 := \cos \left(ky + ky\right)\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot t\_3\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.4:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \frac{1}{\sqrt{1 - 0.5 \cdot \left(t\_3 + t\_2\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (cos (+ kx kx)))
        (t_3 (cos (+ ky ky))))
   (if (<= t_1 -0.99)
     (* (/ (sin ky) (sqrt (fma kx kx (- 0.5 (* 0.5 t_3))))) (sin th))
     (if (<= t_1 -0.4)
       (* (* th (sin ky)) (/ 1.0 (sqrt (- 1.0 (* 0.5 (+ t_3 t_2))))))
       (if (<= t_1 0.7)
         (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 t_2)))) (sin th))
         (if (<= t_1 2.0) (sin th) (* (/ (sin ky) (sin kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = cos((kx + kx));
	double t_3 = cos((ky + ky));
	double tmp;
	if (t_1 <= -0.99) {
		tmp = (sin(ky) / sqrt(fma(kx, kx, (0.5 - (0.5 * t_3))))) * sin(th);
	} else if (t_1 <= -0.4) {
		tmp = (th * sin(ky)) * (1.0 / sqrt((1.0 - (0.5 * (t_3 + t_2)))));
	} else if (t_1 <= 0.7) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * t_2)))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = cos(Float64(kx + kx))
	t_3 = cos(Float64(ky + ky))
	tmp = 0.0
	if (t_1 <= -0.99)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(kx, kx, Float64(0.5 - Float64(0.5 * t_3))))) * sin(th));
	elseif (t_1 <= -0.4)
		tmp = Float64(Float64(th * sin(ky)) * Float64(1.0 / sqrt(Float64(1.0 - Float64(0.5 * Float64(t_3 + t_2))))));
	elseif (t_1 <= 0.7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * t_2)))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(kx * kx + N[(0.5 - N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.4], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 - N[(0.5 * N[(t$95$3 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 86.2

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 65.3

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 64.9

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

   3. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-sin-a-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-fma.f64 86.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.0

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. pow2 N/A

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

      3. sqr-sin-a-rev N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 62.7

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lift-+.f64 62.7

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

   8. Applied rewrites 62.7%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40000000000000002

   1. Initial program 99.3%

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 99.3

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

   3. Applied rewrites 99.3%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqr-sin-a-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sqr-sin-a-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 N/A

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

      16. +-commutative N/A

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

      17. lift--.f64 N/A

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

      18. associate-+l- N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

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

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

   8. Applied rewrites 50.9%

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

   if -0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.2

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 99.0

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 78.4

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

   3. Applied rewrites 78.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 78.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 78.4

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 78.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.4

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 78.4

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

   5. Applied rewrites 78.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 39.5

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

   7. Applied rewrites 39.5%

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

   8. Taylor expanded in ky around 0

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

      1. pow2 N/A

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

      2. sqr-sin-a-rev N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 63.8

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f64 63.8

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

   10. Applied rewrites 63.8%

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

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.6

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 80.0

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 79.5

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

   3. Applied rewrites 79.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-sin-a-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-fma.f64 99.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.2

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in kx around 0

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

      1. lift-sin.f64 78.7

         \[\leadsto \sin th \]

   8. Applied rewrites 78.7%

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

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

   1. Initial program 2.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 2.6

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 2.6

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 2.6

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

   3. Applied rewrites 2.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 2.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 2.6

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 2.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 2.6

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

      15. lift-*.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 2.6

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

   5. Applied rewrites 2.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. sqr-sin-a-rev N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow-to-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 1.5

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

   7. Applied rewrites 1.5%

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

   8. Taylor expanded in ky around 0

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

      1. lift-sin.f64 29.8

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

   10. Applied rewrites 29.8%

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

Alternative 6: 65.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.4:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.99)
     (*
      (/ (sin ky) (sqrt (fma kx kx (- 0.5 (* 0.5 (cos (+ ky ky)))))))
      (sin th))
     (if (<= t_1 -0.4)
       (* (* th (sin ky)) (/ 1.0 (hypot (sin kx) (sin ky))))
       (if (<= t_1 0.7)
         (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ kx kx)))))) (sin th))
         (if (<= t_1 2.0) (sin th) (* (/ (sin ky) (sin kx)) (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 <= -0.99) {
		tmp = (sin(ky) / sqrt(fma(kx, kx, (0.5 - (0.5 * cos((ky + ky))))))) * sin(th);
	} else if (t_1 <= -0.4) {
		tmp = (th * sin(ky)) * (1.0 / hypot(sin(kx), sin(ky)));
	} else if (t_1 <= 0.7) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((kx + kx)))))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / sin(kx)) * 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 <= -0.99)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(kx, kx, Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky))))))) * sin(th));
	elseif (t_1 <= -0.4)
		tmp = Float64(Float64(th * sin(ky)) * Float64(1.0 / hypot(sin(kx), sin(ky))));
	elseif (t_1 <= 0.7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx)))))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / sin(kx)) * 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, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(kx * kx + N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.4], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 86.2

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

      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. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 65.3

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 64.9

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

   3. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-sin-a-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-fma.f64 86.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.0

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 86.0

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 86.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      3. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right)}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, \color{blue}{kx}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      5. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      8. lift--.f64 62.7

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin th \]

      11. lift-+.f64 62.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot \sin th \]

   8. Applied rewrites 62.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(ky + ky\right)\right)}}} \cdot \sin th \]

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40000000000000002

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.3

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 99.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 99.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 99.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-fma.f64 99.1

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      12. lower-*.f64 99.1

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 99.1

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \color{blue}{\sqrt{\frac{1}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{1}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      3. lift-sin.f64 N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\color{blue}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \frac{1}{\sqrt{\color{blue}{\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      6. pow1/2 N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \frac{1}{{\left(\left(\frac{1}{2} + {\sin ky}^{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}^{\color{blue}{\frac{1}{2}}}} \]

      7. pow2 N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \frac{1}{{\left(\left(\frac{1}{2} + \sin ky \cdot \sin ky\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}^{\frac{1}{2}}} \]

      8. sqr-sin-a-rev N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \frac{1}{{\left(\left(\frac{1}{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}^{\frac{1}{2}}} \]

      9. +-commutative N/A

         \[\leadsto \left(th \cdot \sin ky\right) \cdot \frac{1}{{\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \frac{1}{2}\right) - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}^{\frac{1}{2}}} \]

      10. associate-+r- N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \frac{1}{{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}^{\frac{1}{2}}} \]

      11. metadata-eval N/A

          \[\leadsto \left(th \cdot \sin ky\right) \cdot \frac{1}{{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)}} \]

   8. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   if -0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.2

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 99.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 78.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 78.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 78.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 78.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 78.4

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 78.4

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 78.4

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 78.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 39.5

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 39.5%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      6. lift--.f64 63.8

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      8. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

      9. lift-+.f64 63.8

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

   10. Applied rewrites 63.8%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}} \cdot \sin th \]

   if 0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 80.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 79.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 79.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-fma.f64 99.2

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      12. lower-*.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   7. Step-by-step derivation

      1. lift-sin.f64 78.7

         \[\leadsto \sin th \]

   8. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 2.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 2.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 1.5

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 1.5%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-sin.f64 29.8

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \sin th \]

   10. Applied rewrites 29.8%

       \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 64.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.0115:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, kx \cdot kx\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.0115)
   (* (/ (sin ky) (sqrt (fma (sin ky) (sin ky) (* kx kx)))) (sin th))
   (/ (* (sin th) (sin ky)) (sqrt (- 0.5 (* 0.5 (cos (+ kx kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.0115) {
		tmp = (sin(ky) / sqrt(fma(sin(ky), sin(ky), (kx * kx)))) * sin(th);
	} else {
		tmp = (sin(th) * sin(ky)) / sqrt((0.5 - (0.5 * cos((kx + kx)))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.0115)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(sin(ky), sin(ky), Float64(kx * kx)))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) * sin(ky)) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.0115], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Sin[ky], $MachinePrecision] * N[Sin[ky], $MachinePrecision] + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.0115

   1. Initial program 92.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 92.3

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 79.8

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 67.7

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 67.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-fma.f64 80.3

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      12. lower-*.f64 80.3

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 80.3

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 80.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \color{blue}{{kx}^{2}}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, kx \cdot \color{blue}{kx}\right)}} \cdot \sin th \]

      2. lower-*.f64 65.4

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, kx \cdot \color{blue}{kx}\right)}} \cdot \sin th \]

   8. Applied rewrites 65.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \color{blue}{kx \cdot kx}\right)}} \cdot \sin th \]

   if 0.0115 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lower-*.f64 99.4

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin kx \cdot \sin kx + \color{blue}{{\sin ky}^{2}}}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}} \]

      13. lower-hypot.f64 99.5

          \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   3. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]
   4. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + \sin ky \cdot \sin ky}} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + \sin ky \cdot \sin ky}} \]

      4. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + \sin ky \cdot \sin ky}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right) + \sin ky \cdot \sin ky}} \]

      6. lift-cos.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right) + \sin ky \cdot \sin ky}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right) + \sin ky \cdot \sin ky}} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + \sin ky \cdot \sin ky}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \color{blue}{\sin ky} \cdot \sin ky}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \sin ky \cdot \color{blue}{\sin ky}}} \]

      11. sqr-sin-a-rev N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right)}} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right)}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \]

      16. +-commutative N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \]

      18. associate-+l- N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \]

      4. lift-+.f64 60.0

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \]

   8. Applied rewrites 60.0%

      \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(kx + kx\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.7:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.7)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th))
     (if (<= t_1 0.7)
       (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ kx kx)))))) (sin th))
       (if (<= t_1 2.0) (sin th) (* (/ (sin ky) (sin kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.7) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	} else if (t_1 <= 0.7) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((kx + kx)))))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    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 <= (-0.7d0)) then
        tmp = (sin(ky) / sqrt((0.5d0 - (0.5d0 * cos((ky + ky)))))) * sin(th)
    else if (t_1 <= 0.7d0) then
        tmp = (sin(ky) / sqrt((0.5d0 - (0.5d0 * cos((kx + kx)))))) * sin(th)
    else if (t_1 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = (sin(ky) / sin(kx)) * 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 <= -0.7) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((ky + ky)))))) * Math.sin(th);
	} else if (t_1 <= 0.7) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((kx + kx)))))) * Math.sin(th);
	} else if (t_1 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / Math.sin(kx)) * 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 <= -0.7:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((ky + ky)))))) * math.sin(th)
	elif t_1 <= 0.7:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((kx + kx)))))) * math.sin(th)
	elif t_1 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(ky) / math.sin(kx)) * 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 <= -0.7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	elseif (t_1 <= 0.7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx)))))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / sin(kx)) * 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 <= -0.7)
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	elseif (t_1 <= 0.7)
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((kx + kx)))))) * sin(th);
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = (sin(ky) / sin(kx)) * 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, -0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.69999999999999996

   1. Initial program 89.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 89.3

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 73.2

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 72.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 72.9

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 72.9

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 72.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 36.0

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 36.0%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      3. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot \sin th \]

      4. lift-+.f64 53.2

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th \]

   10. Applied rewrites 53.2%

       \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(ky + ky\right)}}} \cdot \sin th \]

   if -0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.2

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 99.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 80.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 80.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 80.0

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 80.0

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 80.0

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 80.0

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 80.0

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 80.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 40.5

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 40.5%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      6. lift--.f64 60.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      8. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

      9. lift-+.f64 60.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

   10. Applied rewrites 60.4%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}} \cdot \sin th \]

   if 0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 80.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 79.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 79.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-fma.f64 99.2

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      12. lower-*.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   7. Step-by-step derivation

      1. lift-sin.f64 78.7

         \[\leadsto \sin th \]

   8. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 2.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 2.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 1.5

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 1.5%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-sin.f64 29.8

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \sin th \]

   10. Applied rewrites 29.8%

       \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 49.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 1.05 \cdot 10^{-169}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;ky \leq 0.009:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 1.05e-169)
   (* (/ ky (sin kx)) (sin th))
   (if (<= ky 0.009)
     (*
      (/ (sin ky) (sqrt (fma ky ky (- 0.5 (* 0.5 (cos (+ kx kx)))))))
      (sin th))
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 1.05e-169) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (ky <= 0.009) {
		tmp = (sin(ky) / sqrt(fma(ky, ky, (0.5 - (0.5 * cos((kx + kx))))))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 1.05e-169)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (ky <= 0.009)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(ky, ky, Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx))))))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 1.05e-169], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 0.009], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * ky + N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 1.05e-169

   1. Initial program 90.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 90.7

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 83.3

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 71.2

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 71.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 71.2

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 71.2

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 71.2

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 71.2

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 71.2

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 71.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 34.8

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 34.8%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 31.0

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   10. Applied rewrites 31.0%

       \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.05e-169 < ky < 0.00899999999999999932

   1. Initial program 98.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 98.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 66.3

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 53.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 53.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 53.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 53.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 53.4

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 53.4

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 53.4

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 53.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 29.7

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 29.7%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{ky \cdot ky + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{ky \cdot ky + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      3. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{ky \cdot ky + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, \color{blue}{ky}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift--.f64 85.5

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

      11. lift-+.f64 85.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

   10. Applied rewrites 85.5%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(ky, ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}}} \cdot \sin th \]

   if 0.00899999999999999932 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 99.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 98.9

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 98.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 98.9

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 98.9

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 98.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 98.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 98.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 48.1

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 48.1%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      3. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot \sin th \]

      4. lift-+.f64 59.5

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th \]

   10. Applied rewrites 59.5%

       \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(ky + ky\right)}}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 47.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 0.7)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ kx kx)))))) (sin th))
     (if (<= t_1 2.0) (sin th) (* (/ (sin ky) (sin kx)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 0.7) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((kx + kx)))))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    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 <= 0.7d0) then
        tmp = (sin(ky) / sqrt((0.5d0 - (0.5d0 * cos((kx + kx)))))) * sin(th)
    else if (t_1 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = (sin(ky) / sin(kx)) * 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 <= 0.7) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((kx + kx)))))) * Math.sin(th);
	} else if (t_1 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / Math.sin(kx)) * 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 <= 0.7:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((kx + kx)))))) * math.sin(th)
	elif t_1 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(ky) / math.sin(kx)) * 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 <= 0.7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx)))))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / sin(kx)) * 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 <= 0.7)
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((kx + kx)))))) * sin(th);
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = (sin(ky) / sin(kx)) * 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, 0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 95.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 95.5

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 89.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 77.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 77.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 77.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 77.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 77.4

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 77.4

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 77.4

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 77.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 38.8

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 38.8%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      6. lift--.f64 40.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      8. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

      9. lift-+.f64 40.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

   10. Applied rewrites 40.6%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}} \cdot \sin th \]

   if 0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 80.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 79.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 79.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-fma.f64 99.2

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      12. lower-*.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   7. Step-by-step derivation

      1. lift-sin.f64 78.7

         \[\leadsto \sin th \]

   8. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 2.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 2.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 1.5

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 1.5%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-sin.f64 29.8

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \sin th \]

   10. Applied rewrites 29.8%

       \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 46.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.135:\\ \;\;\;\;\left(ky \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}\right) \cdot \sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.135)
   (* (* ky (/ 1.0 (sqrt (- 0.5 (* 0.5 (cos (+ kx kx))))))) (sin th))
   (if (<= (sin kx) 5e-27) (sin th) (* (/ (sin ky) (sin kx)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.135) {
		tmp = (ky * (1.0 / sqrt((0.5 - (0.5 * cos((kx + kx))))))) * sin(th);
	} else if (sin(kx) <= 5e-27) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.135d0)) then
        tmp = (ky * (1.0d0 / sqrt((0.5d0 - (0.5d0 * cos((kx + kx))))))) * sin(th)
    else if (sin(kx) <= 5d-27) then
        tmp = sin(th)
    else
        tmp = (sin(ky) / sin(kx)) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.135) {
		tmp = (ky * (1.0 / Math.sqrt((0.5 - (0.5 * Math.cos((kx + kx))))))) * Math.sin(th);
	} else if (Math.sin(kx) <= 5e-27) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.135:
		tmp = (ky * (1.0 / math.sqrt((0.5 - (0.5 * math.cos((kx + kx))))))) * math.sin(th)
	elif math.sin(kx) <= 5e-27:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.135)
		tmp = Float64(Float64(ky * Float64(1.0 / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx))))))) * sin(th));
	elseif (sin(kx) <= 5e-27)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.135)
		tmp = (ky * (1.0 / sqrt((0.5 - (0.5 * cos((kx + kx))))))) * sin(th);
	elseif (sin(kx) <= 5e-27)
		tmp = sin(th);
	else
		tmp = (sin(ky) / sin(kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.135], N[(N[(ky * N[(1.0 / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-27], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.13500000000000001

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. pow1/2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\frac{1}{2}}}} \cdot \sin th \]

      3. sqr-pow N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}} \cdot \sin th \]

   3. Applied rewrites 99.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}^{0.25} \cdot {\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}^{0.25}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(ky \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      2. sqrt-div N/A

         \[\leadsto \left(ky \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      6. lift-cos.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      8. lift-*.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      9. lift--.f64 50.0

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}}\right) \cdot \sin th \]

      12. lift-+.f64 50.0

          \[\leadsto \left(ky \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}\right) \cdot \sin th \]

   6. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\left(ky \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}\right)} \cdot \sin th \]

   if -0.13500000000000001 < (sin.f64 kx) < 5.0000000000000002e-27

   1. Initial program 88.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 88.9

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 70.7

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 54.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 54.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-fma.f64 72.9

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      12. lower-*.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 72.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   7. Step-by-step derivation

      1. lift-sin.f64 38.8

         \[\leadsto \sin th \]

   8. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\sin th} \]

   if 5.0000000000000002e-27 < (sin.f64 kx)

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.4

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 96.1

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 96.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 96.1

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 96.1

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 96.1

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 96.1

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 96.1

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 96.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 96.1

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 96.1%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-sin.f64 60.3

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \sin th \]

   10. Applied rewrites 60.3%

       \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 46.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(ky \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 5e-5)
     (* (* ky (/ 1.0 (sqrt (- 0.5 (* 0.5 (cos (+ kx kx))))))) (sin th))
     (if (<= t_1 2.0) (sin th) (* (/ ky (sin kx)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (ky * (1.0 / sqrt((0.5 - (0.5 * cos((kx + kx))))))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / sin(kx)) * sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= 5d-5) then
        tmp = (ky * (1.0d0 / sqrt((0.5d0 - (0.5d0 * cos((kx + kx))))))) * sin(th)
    else if (t_1 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = (ky / sin(kx)) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (ky * (1.0 / Math.sqrt((0.5 - (0.5 * Math.cos((kx + kx))))))) * Math.sin(th);
	} else if (t_1 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 5e-5:
		tmp = (ky * (1.0 / math.sqrt((0.5 - (0.5 * math.cos((kx + kx))))))) * math.sin(th)
	elif t_1 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = Float64(Float64(ky * Float64(1.0 / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx))))))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 5e-5)
		tmp = (ky * (1.0 / sqrt((0.5 - (0.5 * cos((kx + kx))))))) * sin(th);
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = (ky / sin(kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], N[(N[(ky * N[(1.0 / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.00000000000000024e-5

   1. Initial program 95.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. pow1/2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\frac{1}{2}}}} \cdot \sin th \]

      3. sqr-pow N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}} \cdot \sin th \]

   3. Applied rewrites 75.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}^{0.25} \cdot {\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}^{0.25}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(ky \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      2. sqrt-div N/A

         \[\leadsto \left(ky \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      6. lift-cos.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      7. lift-*.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      8. lift-*.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      9. lift--.f64 38.4

         \[\leadsto \left(ky \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \left(ky \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}}\right) \cdot \sin th \]

      12. lift-+.f64 38.4

          \[\leadsto \left(ky \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}\right) \cdot \sin th \]

   6. Applied rewrites 38.4%

      \[\leadsto \color{blue}{\left(ky \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}\right)} \cdot \sin th \]

   if 5.00000000000000024e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.5

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 83.7

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 83.3

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 83.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-fma.f64 99.1

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      12. lower-*.f64 99.1

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 99.1

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   7. Step-by-step derivation

      1. lift-sin.f64 66.6

         \[\leadsto \sin th \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 2.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 2.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 1.5

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 1.5%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 29.8

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   10. Applied rewrites 29.8%

       \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 46.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 5e-5)
     (* (* ky (sin th)) (/ 1.0 (sqrt (- 0.5 (* 0.5 (cos (+ kx kx)))))))
     (if (<= t_1 2.0) (sin th) (* (/ ky (sin kx)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (ky * sin(th)) * (1.0 / sqrt((0.5 - (0.5 * cos((kx + kx))))));
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / sin(kx)) * sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= 5d-5) then
        tmp = (ky * sin(th)) * (1.0d0 / sqrt((0.5d0 - (0.5d0 * cos((kx + kx))))))
    else if (t_1 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = (ky / sin(kx)) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-5) {
		tmp = (ky * Math.sin(th)) * (1.0 / Math.sqrt((0.5 - (0.5 * Math.cos((kx + kx))))));
	} else if (t_1 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 5e-5:
		tmp = (ky * math.sin(th)) * (1.0 / math.sqrt((0.5 - (0.5 * math.cos((kx + kx))))))
	elif t_1 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 5e-5)
		tmp = Float64(Float64(ky * sin(th)) * Float64(1.0 / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx)))))));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 5e-5)
		tmp = (ky * sin(th)) * (1.0 / sqrt((0.5 - (0.5 * cos((kx + kx))))));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = (ky / sin(kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-5], N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.00000000000000024e-5

   1. Initial program 95.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lower-*.f64 93.2

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin kx \cdot \sin kx + \color{blue}{{\sin ky}^{2}}}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}} \]

      13. lower-hypot.f64 96.3

          \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   3. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]
   4. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + \sin ky \cdot \sin ky}} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + \sin ky \cdot \sin ky}} \]

      4. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + \sin ky \cdot \sin ky}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right) + \sin ky \cdot \sin ky}} \]

      6. lift-cos.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right) + \sin ky \cdot \sin ky}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right) + \sin ky \cdot \sin ky}} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + \sin ky \cdot \sin ky}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \color{blue}{\sin ky} \cdot \sin ky}} \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \sin ky \cdot \color{blue}{\sin ky}}} \]

      11. sqr-sin-a-rev N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right)}} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right)}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \]

      16. +-commutative N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \]

      18. associate-+l- N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \]

   5. Applied rewrites 75.4%

      \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      3. lift-sin.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      11. lift--.f64 38.3

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      13. count-2-rev N/A

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \]

      14. lift-+.f64 38.3

          \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \]

   8. Applied rewrites 38.3%

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \frac{1}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}}} \]

   if 5.00000000000000024e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.5

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 83.7

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 83.3

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 83.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-fma.f64 99.1

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      12. lower-*.f64 99.1

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 99.1

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   7. Step-by-step derivation

      1. lift-sin.f64 66.6

         \[\leadsto \sin th \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 2.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 2.6

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 2.6

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 2.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 1.5

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 1.5%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 29.8

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   10. Applied rewrites 29.8%

       \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 43.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{ky}{\sin kx} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ ky (sin kx)) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 0.04) t_1 (if (<= t_2 2.0) (sin th) t_1))))

C

double code(double kx, double ky, double th) {
	double t_1 = (ky / sin(kx)) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 0.04) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (ky / sin(kx)) * sin(th)
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_2 <= 0.04d0) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (ky / Math.sin(kx)) * Math.sin(th);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 0.04) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (ky / math.sin(kx)) * math.sin(th)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= 0.04:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(ky / sin(kx)) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 0.04)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (ky / sin(kx)) * sin(th);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= 0.04)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.04], t$95$1, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0400000000000000008 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 91.9

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 85.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 72.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 72.9

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 72.9

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 72.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 36.7

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 36.7%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 34.1

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   10. Applied rewrites 34.1%

       \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.5

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 83.7

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 83.3

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 83.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-fma.f64 99.2

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      12. lower-*.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   7. Step-by-step derivation

      1. lift-sin.f64 67.1

         \[\leadsto \sin th \]

   8. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 43.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{ky \cdot \sin th}{\sin kx}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 0.04:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* ky (sin th)) (sin kx)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 0.04) t_1 (if (<= t_2 2.0) (sin th) t_1))))

C

double code(double kx, double ky, double th) {
	double t_1 = (ky * sin(th)) / sin(kx);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 0.04) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (ky * sin(th)) / sin(kx)
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_2 <= 0.04d0) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (ky * Math.sin(th)) / Math.sin(kx);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 0.04) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (ky * math.sin(th)) / math.sin(kx)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= 0.04:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(ky * sin(th)) / sin(kx))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 0.04)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (ky * sin(th)) / sin(kx);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= 0.04)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(ky * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.04], t$95$1, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0400000000000000008 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 91.9

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 85.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 72.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. associate-+l- N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right) - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      5. lower--.f64 72.9

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\left(0.5 \cdot \cos \left(2 \cdot ky\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 72.9

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5} - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      10. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      11. lower-+.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \color{blue}{\left(ky + ky\right)} \cdot 0.5 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      13. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)\right)}} \cdot \sin th \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)\right)}} \cdot \sin th \]

      17. lower-+.f64 72.9

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 72.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}\right)}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right)\right)}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right)\right)}} \cdot \sin th \]

      6. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}} \cdot \sin th \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{\sin kx} \cdot \sin kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \sin kx \cdot \color{blue}{\sin kx}\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{{\sin kx}^{2}}\right)}} \cdot \sin th \]

      11. pow-to-exp N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \left(\cos \left(ky + ky\right) \cdot \frac{1}{2} - e^{\color{blue}{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

      14. lower-log.f64 36.7

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - e^{\color{blue}{\log \sin kx} \cdot 2}\right)}} \cdot \sin th \]

   7. Applied rewrites 36.7%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \color{blue}{e^{\log \sin kx \cdot 2}}\right)}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sin kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sin \color{blue}{kx}} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sin kx} \]

      4. lift-sin.f64 33.1

         \[\leadsto \frac{ky \cdot \sin th}{\sin kx} \]

   10. Applied rewrites 33.1%

       \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

   if 0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 99.5

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 83.7

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 83.3

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 83.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      9. lift-fma.f64 99.2

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      12. lower-*.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      14. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower-+.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   7. Step-by-step derivation

      1. lift-sin.f64 67.1

         \[\leadsto \sin th \]

   8. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 23.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (sin th))

C

double code(double kx, double ky, double th) {
	return sin(th);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th);
}

Python

def code(kx, ky, th):
	return math.sin(th)

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   2. +-commutative N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

   3. lower-+.f64 94.1

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

   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. sqr-sin-a N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

   11. lower-cos.f64 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

   12. lower-*.f64 85.0

       \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

   14. unpow2 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

   15. lift-sin.f64 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

   16. lift-sin.f64 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

   17. sqr-sin-a N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   18. lower--.f64 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   19. lower-*.f64 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   20. lower-cos.f64 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   21. lower-*.f64 76.0

       \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

3. Applied rewrites 76.0%

   \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

   4. lift-cos.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

   6. sqr-sin-a-rev N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

   7. lift-sin.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

   8. lift-sin.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

   9. lift-fma.f64 85.3

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   11. *-commutative N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

   12. lower-*.f64 85.3

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \color{blue}{\cos \left(2 \cdot kx\right) \cdot 0.5}\right)}} \cdot \sin th \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

   14. count-2-rev N/A

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, \frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

   15. lower-+.f64 85.3

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \color{blue}{\left(kx + kx\right)} \cdot 0.5\right)}} \cdot \sin th \]

5. Applied rewrites 85.3%

   \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, 0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)}}} \cdot \sin th \]

6. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
7. Step-by-step derivation

   1. lift-sin.f64 23.7

      \[\leadsto \sin th \]

8. Applied rewrites 23.7%

   \[\leadsto \color{blue}{\sin th} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(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)))