Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.3% → 99.7%

Time: 11.8s

Alternatives: 30

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: 94.3% 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 92.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 67.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.848:\\ \;\;\;\;\left(t\_1 \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (sqrt
          (pow
           (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0)
           -1.0)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.995)
     (*
      (*
       (sqrt
        (/
         2.0
         (-
          (fma (fma -0.6666666666666666 (* kx kx) 2.0) (* kx kx) 1.0)
          (cos (* -2.0 ky)))))
       (sin ky))
      (sin th))
     (if (<= t_2 -0.32)
       (* (* (sin ky) th) t_1)
       (if (<= t_2 5e-201)
         (* (* (sqrt (/ 2.0 (- 1.0 (cos (* -2.0 kx))))) (sin ky)) (sin th))
         (if (<= t_2 0.02)
           (* (/ (sin ky) (sin kx)) (sin th))
           (if (<= t_2 0.848)
             (*
              (* t_1 (sin ky))
              (* (fma (* th th) -0.16666666666666666 1.0) th))
             (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.995) {
		tmp = (sqrt((2.0 / (fma(fma(-0.6666666666666666, (kx * kx), 2.0), (kx * kx), 1.0) - cos((-2.0 * ky))))) * sin(ky)) * sin(th);
	} else if (t_2 <= -0.32) {
		tmp = (sin(ky) * th) * t_1;
	} else if (t_2 <= 5e-201) {
		tmp = (sqrt((2.0 / (1.0 - cos((-2.0 * kx))))) * sin(ky)) * sin(th);
	} else if (t_2 <= 0.02) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_2 <= 0.848) {
		tmp = (t_1 * sin(ky)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.995)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(fma(fma(-0.6666666666666666, Float64(kx * kx), 2.0), Float64(kx * kx), 1.0) - cos(Float64(-2.0 * ky))))) * sin(ky)) * sin(th));
	elseif (t_2 <= -0.32)
		tmp = Float64(Float64(sin(ky) * th) * t_1);
	elseif (t_2 <= 5e-201)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * sin(ky)) * sin(th));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_2 <= 0.848)
		tmp = Float64(Float64(t_1 * sin(ky)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Power[N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.995], N[(N[(N[Sqrt[N[(2.0 / N[(N[(N[(-0.6666666666666666 * N[(kx * kx), $MachinePrecision] + 2.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.32], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 5e-201], N[(N[(N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.848], N[(N[(t$95$1 * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 84.0%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 68.3%

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

   11. Taylor expanded in kx around 0

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

   13. Applied rewrites 66.4%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.320000000000000007

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -0.320000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 69.6%

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

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.847999999999999976

   1. Initial program 99.3%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.3

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

   10. Applied rewrites 65.3%

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

   if 0.847999999999999976 < (/.f64 (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 81.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 88.3

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

   5. Applied rewrites 88.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 6 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.848:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}} \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.32:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.848:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (* (sin ky) th)
          (sqrt
           (pow
            (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0)
            -1.0))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.995)
     (*
      (*
       (sqrt
        (/
         2.0
         (-
          (fma (fma -0.6666666666666666 (* kx kx) 2.0) (* kx kx) 1.0)
          (cos (* -2.0 ky)))))
       (sin ky))
      (sin th))
     (if (<= t_2 -0.32)
       t_1
       (if (<= t_2 5e-201)
         (* (* (sqrt (/ 2.0 (- 1.0 (cos (* -2.0 kx))))) (sin ky)) (sin th))
         (if (<= t_2 0.02)
           (* (/ (sin ky) (sin kx)) (sin th))
           (if (<= t_2 0.848) t_1 (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) * sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.995) {
		tmp = (sqrt((2.0 / (fma(fma(-0.6666666666666666, (kx * kx), 2.0), (kx * kx), 1.0) - cos((-2.0 * ky))))) * sin(ky)) * sin(th);
	} else if (t_2 <= -0.32) {
		tmp = t_1;
	} else if (t_2 <= 5e-201) {
		tmp = (sqrt((2.0 / (1.0 - cos((-2.0 * kx))))) * sin(ky)) * sin(th);
	} else if (t_2 <= 0.02) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_2 <= 0.848) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) * sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.995)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(fma(fma(-0.6666666666666666, Float64(kx * kx), 2.0), Float64(kx * kx), 1.0) - cos(Float64(-2.0 * ky))))) * sin(ky)) * sin(th));
	elseif (t_2 <= -0.32)
		tmp = t_1;
	elseif (t_2 <= 5e-201)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * sin(ky)) * sin(th));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_2 <= 0.848)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.995], N[(N[(N[Sqrt[N[(2.0 / N[(N[(N[(-0.6666666666666666 * N[(kx * kx), $MachinePrecision] + 2.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.32], t$95$1, If[LessEqual[t$95$2, 5e-201], N[(N[(N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.848], t$95$1, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 84.0%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 68.3%

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

   11. Taylor expanded in kx around 0

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

   13. Applied rewrites 66.4%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.320000000000000007 or 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.847999999999999976

   1. Initial program 99.4%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.8%

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

   if -0.320000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 69.6%

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

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   if 0.847999999999999976 < (/.f64 (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 81.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.848:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.32:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.848:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (* (sin ky) th)
          (sqrt
           (pow
            (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0)
            -1.0))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.995)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (fma -0.5 (cos (* -2.0 ky)) 0.5))))
      (sin th))
     (if (<= t_2 -0.32)
       t_1
       (if (<= t_2 5e-201)
         (* (* (sqrt (/ 2.0 (- 1.0 (cos (* -2.0 kx))))) (sin ky)) (sin th))
         (if (<= t_2 0.02)
           (* (/ (sin ky) (sin kx)) (sin th))
           (if (<= t_2 0.848) t_1 (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) * sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.995) {
		tmp = (sin(ky) / sqrt(((kx * kx) + fma(-0.5, cos((-2.0 * ky)), 0.5)))) * sin(th);
	} else if (t_2 <= -0.32) {
		tmp = t_1;
	} else if (t_2 <= 5e-201) {
		tmp = (sqrt((2.0 / (1.0 - cos((-2.0 * kx))))) * sin(ky)) * sin(th);
	} else if (t_2 <= 0.02) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_2 <= 0.848) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) * sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.995)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + fma(-0.5, cos(Float64(-2.0 * ky)), 0.5)))) * sin(th));
	elseif (t_2 <= -0.32)
		tmp = t_1;
	elseif (t_2 <= 5e-201)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * sin(ky)) * sin(th));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_2 <= 0.848)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(-0.5 * N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.32], t$95$1, If[LessEqual[t$95$2, 5e-201], N[(N[(N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.848], t$95$1, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 84.0%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. cos-2 N/A

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

      10. cos-sum N/A

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

      11. lower-cos.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-*.f64 66.7

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

   7. Applied rewrites 66.7%

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

   8. Taylor expanded in ky around inf

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. cos-neg-rev N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower-cos.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. lower-*.f64 66.7

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

   10. Applied rewrites 66.7%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.320000000000000007 or 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.847999999999999976

   1. Initial program 99.4%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.8%

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

   if -0.320000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 69.6%

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

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   if 0.847999999999999976 < (/.f64 (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 81.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.848:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ t_2 := \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq 0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.848:\\ \;\;\;\;\left(t\_2 \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (/
           (sin ky)
           (hypot (* (fma (* ky ky) -0.16666666666666666 1.0) ky) (sin kx)))
          (sin th)))
        (t_2
         (sqrt
          (pow
           (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0)
           -1.0)))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.995)
     (*
      (*
       (sqrt
        (/
         2.0
         (-
          (fma (fma -0.6666666666666666 (* kx kx) 2.0) (* kx kx) 1.0)
          (cos (* -2.0 ky)))))
       (sin ky))
      (sin th))
     (if (<= t_3 -0.32)
       (* (* (sin ky) th) t_2)
       (if (<= t_3 0.02)
         t_1
         (if (<= t_3 0.848)
           (* (* t_2 (sin ky)) (* (fma (* th th) -0.16666666666666666 1.0) th))
           (if (<= t_3 1.0) (sin th) t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot((fma((ky * ky), -0.16666666666666666, 1.0) * ky), sin(kx))) * sin(th);
	double t_2 = sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0));
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.995) {
		tmp = (sqrt((2.0 / (fma(fma(-0.6666666666666666, (kx * kx), 2.0), (kx * kx), 1.0) - cos((-2.0 * ky))))) * sin(ky)) * sin(th);
	} else if (t_3 <= -0.32) {
		tmp = (sin(ky) * th) * t_2;
	} else if (t_3 <= 0.02) {
		tmp = t_1;
	} else if (t_3 <= 0.848) {
		tmp = (t_2 * sin(ky)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_3 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky), sin(kx))) * sin(th))
	t_2 = sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0))
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.995)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(fma(fma(-0.6666666666666666, Float64(kx * kx), 2.0), Float64(kx * kx), 1.0) - cos(Float64(-2.0 * ky))))) * sin(ky)) * sin(th));
	elseif (t_3 <= -0.32)
		tmp = Float64(Float64(sin(ky) * th) * t_2);
	elseif (t_3 <= 0.02)
		tmp = t_1;
	elseif (t_3 <= 0.848)
		tmp = Float64(Float64(t_2 * sin(ky)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_3 <= 1.0)
		tmp = sin(th);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Power[N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.995], N[(N[(N[Sqrt[N[(2.0 / N[(N[(N[(-0.6666666666666666 * N[(kx * kx), $MachinePrecision] + 2.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.32], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 0.02], t$95$1, If[LessEqual[t$95$3, 0.848], N[(N[(t$95$2 * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1.0], N[Sin[th], $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 84.0%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 68.3%

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

   11. Taylor expanded in kx around 0

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

   13. Applied rewrites 66.4%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.320000000000000007

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -0.320000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004 or 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 93.8

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

   7. Applied rewrites 93.8%

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

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.847999999999999976

   1. Initial program 99.3%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 65.3

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

   10. Applied rewrites 65.3%

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

   if 0.847999999999999976 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 91.0

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

   5. Applied rewrites 91.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.848:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}} \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.848:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.995)
     (*
      (*
       (sqrt
        (/
         2.0
         (-
          (fma (fma -0.6666666666666666 (* kx kx) 2.0) (* kx kx) 1.0)
          (cos (* -2.0 ky)))))
       (sin ky))
      (sin th))
     (if (<= t_1 -0.32)
       (*
        (* (sin ky) th)
        (sqrt
         (pow
          (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0)
          -1.0)))
       (if (<= t_1 5e-201)
         (* (* (sqrt (/ 2.0 (- 1.0 (cos (* -2.0 kx))))) (sin ky)) (sin th))
         (if (<= t_1 0.0001)
           (* (/ (sin ky) (sin kx)) (sin th))
           (if (<= t_1 0.848)
             (* (/ (sin ky) (hypot (sin kx) (sin ky))) th)
             (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.995) {
		tmp = (sqrt((2.0 / (fma(fma(-0.6666666666666666, (kx * kx), 2.0), (kx * kx), 1.0) - cos((-2.0 * ky))))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.32) {
		tmp = (sin(ky) * th) * sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0));
	} else if (t_1 <= 5e-201) {
		tmp = (sqrt((2.0 / (1.0 - cos((-2.0 * kx))))) * sin(ky)) * sin(th);
	} else if (t_1 <= 0.0001) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.848) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.995)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(fma(fma(-0.6666666666666666, Float64(kx * kx), 2.0), Float64(kx * kx), 1.0) - cos(Float64(-2.0 * ky))))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.32)
		tmp = Float64(Float64(sin(ky) * th) * sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0)));
	elseif (t_1 <= 5e-201)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * sin(ky)) * sin(th));
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.848)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.995], N[(N[(N[Sqrt[N[(2.0 / N[(N[(N[(-0.6666666666666666 * N[(kx * kx), $MachinePrecision] + 2.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.32], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-201], N[(N[(N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.848], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 84.0%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 68.3%

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

   11. Taylor expanded in kx around 0

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

   13. Applied rewrites 66.4%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.320000000000000007

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -0.320000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 69.6%

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

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000005e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   if 1.00000000000000005e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.847999999999999976

   1. Initial program 99.3%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 65.0

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

   5. Applied rewrites 65.0%

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

   7. Applied rewrites 65.0%

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

   9. Applied rewrites 65.0%

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

   if 0.847999999999999976 < (/.f64 (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 81.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 88.3

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

   5. Applied rewrites 88.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 6 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.848:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.0001:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.848:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.995)
     (*
      (*
       (sqrt
        (/
         2.0
         (-
          (fma (fma -0.6666666666666666 (* kx kx) 2.0) (* kx kx) 1.0)
          (cos (* -2.0 ky)))))
       (sin ky))
      (sin th))
     (if (<= t_1 -0.32)
       (*
        (* (sin ky) th)
        (sqrt
         (pow
          (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0)
          -1.0)))
       (if (<= t_1 5e-201)
         (* (* (sqrt (/ 2.0 (- 1.0 (cos (* -2.0 kx))))) (sin ky)) (sin th))
         (if (<= t_1 0.0001)
           (* (/ (sin ky) (sin kx)) (sin th))
           (if (<= t_1 0.848)
             (* (sin ky) (/ th (hypot (sin kx) (sin ky))))
             (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.995) {
		tmp = (sqrt((2.0 / (fma(fma(-0.6666666666666666, (kx * kx), 2.0), (kx * kx), 1.0) - cos((-2.0 * ky))))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.32) {
		tmp = (sin(ky) * th) * sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0));
	} else if (t_1 <= 5e-201) {
		tmp = (sqrt((2.0 / (1.0 - cos((-2.0 * kx))))) * sin(ky)) * sin(th);
	} else if (t_1 <= 0.0001) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.848) {
		tmp = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.995)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(fma(fma(-0.6666666666666666, Float64(kx * kx), 2.0), Float64(kx * kx), 1.0) - cos(Float64(-2.0 * ky))))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.32)
		tmp = Float64(Float64(sin(ky) * th) * sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0)));
	elseif (t_1 <= 5e-201)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * sin(ky)) * sin(th));
	elseif (t_1 <= 0.0001)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.848)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(kx), sin(ky))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.995], N[(N[(N[Sqrt[N[(2.0 / N[(N[(N[(-0.6666666666666666 * N[(kx * kx), $MachinePrecision] + 2.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.32], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-201], N[(N[(N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0001], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.848], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 84.0%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 68.3%

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

   11. Taylor expanded in kx around 0

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

   13. Applied rewrites 66.4%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.320000000000000007

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -0.320000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 69.6%

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

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000005e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   if 1.00000000000000005e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.847999999999999976

   1. Initial program 99.3%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 65.0

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

   5. Applied rewrites 65.0%

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

   7. Applied rewrites 65.0%

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

   9. Applied rewrites 65.0%

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

   if 0.847999999999999976 < (/.f64 (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 81.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 88.3

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

   5. Applied rewrites 88.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 6 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.848:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \cos \left(-2 \cdot kx\right)\\ t_3 := \cos \left(-2 \cdot ky\right)\\ \mathbf{if}\;t\_1 \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - t\_3}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - t\_2}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.848:\\ \;\;\;\;\left(\sqrt{\frac{2}{\left(2 - t\_3\right) - t\_2}} \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (cos (* -2.0 kx)))
        (t_3 (cos (* -2.0 ky))))
   (if (<= t_1 -0.995)
     (*
      (*
       (sqrt
        (/
         2.0
         (- (fma (fma -0.6666666666666666 (* kx kx) 2.0) (* kx kx) 1.0) t_3)))
       (sin ky))
      (sin th))
     (if (<= t_1 -0.32)
       (*
        (* (sin ky) th)
        (sqrt
         (pow
          (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0)
          -1.0)))
       (if (<= t_1 5e-201)
         (* (* (sqrt (/ 2.0 (- 1.0 t_2))) (sin ky)) (sin th))
         (if (<= t_1 0.02)
           (* (/ (sin ky) (sin kx)) (sin th))
           (if (<= t_1 0.848)
             (*
              (* (sqrt (/ 2.0 (- (- 2.0 t_3) t_2))) (sin ky))
              (* (fma (* th th) -0.16666666666666666 1.0) th))
             (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = cos((-2.0 * kx));
	double t_3 = cos((-2.0 * ky));
	double tmp;
	if (t_1 <= -0.995) {
		tmp = (sqrt((2.0 / (fma(fma(-0.6666666666666666, (kx * kx), 2.0), (kx * kx), 1.0) - t_3))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.32) {
		tmp = (sin(ky) * th) * sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0));
	} else if (t_1 <= 5e-201) {
		tmp = (sqrt((2.0 / (1.0 - t_2))) * sin(ky)) * sin(th);
	} else if (t_1 <= 0.02) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.848) {
		tmp = (sqrt((2.0 / ((2.0 - t_3) - t_2))) * sin(ky)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = cos(Float64(-2.0 * kx))
	t_3 = cos(Float64(-2.0 * ky))
	tmp = 0.0
	if (t_1 <= -0.995)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(fma(fma(-0.6666666666666666, Float64(kx * kx), 2.0), Float64(kx * kx), 1.0) - t_3))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.32)
		tmp = Float64(Float64(sin(ky) * th) * sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0)));
	elseif (t_1 <= 5e-201)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(1.0 - t_2))) * sin(ky)) * sin(th));
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.848)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(Float64(2.0 - t_3) - t_2))) * sin(ky)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.995], N[(N[(N[Sqrt[N[(2.0 / N[(N[(N[(-0.6666666666666666 * N[(kx * kx), $MachinePrecision] + 2.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.32], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-201], N[(N[(N[Sqrt[N[(2.0 / N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.848], N[(N[(N[Sqrt[N[(2.0 / N[(N[(2.0 - t$95$3), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 84.0%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 68.3%

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

   11. Taylor expanded in kx around 0

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

   13. Applied rewrites 66.4%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.320000000000000007

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -0.320000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 99.6%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 69.6%

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

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.847999999999999976

   1. Initial program 99.3%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 99.2%

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

   11. Taylor expanded in th around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 65.2

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

   13. Applied rewrites 65.2%

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

   if 0.847999999999999976 < (/.f64 (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 81.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 88.3

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

   5. Applied rewrites 88.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 6 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.848:\\ \;\;\;\;\left(\sqrt{\frac{2}{\left(2 - \cos \left(-2 \cdot ky\right)\right) - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ t_4 := \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{if}\;t\_3 \leq -0.995:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot t\_4\\ \mathbf{elif}\;t\_3 \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2 + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.99:\\ \;\;\;\;\left(t\_4 \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (/
           (sin ky)
           (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
          (sin th)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0)))))
        (t_4
         (sqrt
          (pow
           (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0)
           -1.0))))
   (if (<= t_3 -0.995)
     t_1
     (if (<= t_3 -0.32)
       (* (* (sin ky) th) t_4)
       (if (<= t_3 0.02)
         (* (/ (sin ky) (sqrt (+ t_2 (* ky ky)))) (sin th))
         (if (<= t_3 0.99)
           (* (* t_4 (sin ky)) (* (fma (* th th) -0.16666666666666666 1.0) th))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double t_4 = sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0));
	double tmp;
	if (t_3 <= -0.995) {
		tmp = t_1;
	} else if (t_3 <= -0.32) {
		tmp = (sin(ky) * th) * t_4;
	} else if (t_3 <= 0.02) {
		tmp = (sin(ky) / sqrt((t_2 + (ky * ky)))) * sin(th);
	} else if (t_3 <= 0.99) {
		tmp = (t_4 * sin(ky)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	t_4 = sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0))
	tmp = 0.0
	if (t_3 <= -0.995)
		tmp = t_1;
	elseif (t_3 <= -0.32)
		tmp = Float64(Float64(sin(ky) * th) * t_4);
	elseif (t_3 <= 0.02)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(t_2 + Float64(ky * ky)))) * sin(th));
	elseif (t_3 <= 0.99)
		tmp = Float64(Float64(t_4 * sin(ky)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Power[N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -0.995], t$95$1, If[LessEqual[t$95$3, -0.32], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.99], N[(N[(t$95$4 * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.994999999999999996 or 0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.320000000000000007

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -0.320000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.98999999999999999

   1. Initial program 99.2%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.2

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

   10. Applied rewrites 57.2%

       \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.99:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}} \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{if}\;t\_2 \leq -0.995:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot t\_3\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.99:\\ \;\;\;\;\left(t\_3 \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (/
           (sin ky)
           (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
          (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3
         (sqrt
          (pow
           (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0)
           -1.0))))
   (if (<= t_2 -0.995)
     t_1
     (if (<= t_2 -0.32)
       (* (* (sin ky) th) t_3)
       (if (<= t_2 0.02)
         (*
          (/
           (sin ky)
           (hypot (* (fma (* ky ky) -0.16666666666666666 1.0) ky) (sin kx)))
          (sin th))
         (if (<= t_2 0.99)
           (* (* t_3 (sin ky)) (* (fma (* th th) -0.16666666666666666 1.0) th))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0));
	double tmp;
	if (t_2 <= -0.995) {
		tmp = t_1;
	} else if (t_2 <= -0.32) {
		tmp = (sin(ky) * th) * t_3;
	} else if (t_2 <= 0.02) {
		tmp = (sin(ky) / hypot((fma((ky * ky), -0.16666666666666666, 1.0) * ky), sin(kx))) * sin(th);
	} else if (t_2 <= 0.99) {
		tmp = (t_3 * sin(ky)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0))
	tmp = 0.0
	if (t_2 <= -0.995)
		tmp = t_1;
	elseif (t_2 <= -0.32)
		tmp = Float64(Float64(sin(ky) * th) * t_3);
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(sin(ky) / hypot(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky), sin(kx))) * sin(th));
	elseif (t_2 <= 0.99)
		tmp = Float64(Float64(t_3 * sin(ky)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Power[N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -0.995], t$95$1, If[LessEqual[t$95$2, -0.32], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.99], N[(N[(t$95$3 * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.994999999999999996 or 0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.320000000000000007

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -0.320000000000000007 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 93.0

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

   7. Applied rewrites 93.0%

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

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.98999999999999999

   1. Initial program 99.2%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.2

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

   10. Applied rewrites 57.2%

       \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.32:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.99:\\ \;\;\;\;\left(\sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}} \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;t\_1 \leq 0.848:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.995)
     (*
      (*
       (sqrt
        (/
         2.0
         (-
          (fma (fma -0.6666666666666666 (* kx kx) 2.0) (* kx kx) 1.0)
          (cos (* -2.0 ky)))))
       (sin ky))
      (sin th))
     (if (<= t_1 -0.1)
       (*
        (* (sin ky) th)
        (sqrt
         (pow
          (/ (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))) 2.0)
          -1.0)))
       (if (<= t_1 5e-7)
         (/ (* (sin th) ky) (hypot (sin ky) (sin kx)))
         (if (<= t_1 0.848)
           (* (/ (sin ky) (hypot (sin kx) (sin ky))) th)
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.995) {
		tmp = (sqrt((2.0 / (fma(fma(-0.6666666666666666, (kx * kx), 2.0), (kx * kx), 1.0) - cos((-2.0 * ky))))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = (sin(ky) * th) * sqrt(pow((((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))) / 2.0), -1.0));
	} else if (t_1 <= 5e-7) {
		tmp = (sin(th) * ky) / hypot(sin(ky), sin(kx));
	} else if (t_1 <= 0.848) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.995)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(fma(fma(-0.6666666666666666, Float64(kx * kx), 2.0), Float64(kx * kx), 1.0) - cos(Float64(-2.0 * ky))))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = Float64(Float64(sin(ky) * th) * sqrt((Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))) / 2.0) ^ -1.0)));
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(sin(th) * ky) / hypot(sin(ky), sin(kx)));
	elseif (t_1 <= 0.848)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.995], N[(N[(N[Sqrt[N[(2.0 / N[(N[(N[(-0.6666666666666666 * N[(kx * kx), $MachinePrecision] + 2.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.848], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 84.0%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 82.1

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

   5. Applied rewrites 82.1%

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in kx around inf

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

   10. Applied rewrites 68.3%

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

   11. Taylor expanded in kx around 0

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

   13. Applied rewrites 66.4%

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

   if -0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.5%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 53.1

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

   5. Applied rewrites 53.1%

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

   7. Applied rewrites 53.2%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.99999999999999977e-7

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. 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 97.3

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

      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. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 95.6

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

   7. Applied rewrites 95.6%

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

   if 4.99999999999999977e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.847999999999999976

   1. Initial program 99.3%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 63.3

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

   5. Applied rewrites 63.3%

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

   7. Applied rewrites 66.0%

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

   9. Applied rewrites 66.0%

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

   if 0.847999999999999976 < (/.f64 (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 81.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.995:\\ \;\;\;\;\left(\sqrt{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, kx \cdot kx, 2\right), kx \cdot kx, 1\right) - \cos \left(-2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left(\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.848:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.72:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.72)
     (* (/ (sin ky) (sqrt (fma -0.5 (cos (* -2.0 ky)) 0.5))) (sin th))
     (if (<= t_1 5e-201)
       (* (* (sqrt (/ 2.0 (- 1.0 (cos (* -2.0 kx))))) (sin ky)) (sin th))
       (if (<= t_1 0.02) (* (/ (sin ky) (sin kx)) (sin th)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.72) {
		tmp = (sin(ky) / sqrt(fma(-0.5, cos((-2.0 * ky)), 0.5))) * sin(th);
	} else if (t_1 <= 5e-201) {
		tmp = (sqrt((2.0 / (1.0 - cos((-2.0 * kx))))) * sin(ky)) * sin(th);
	} else if (t_1 <= 0.02) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.72)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(-2.0 * ky)), 0.5))) * sin(th));
	elseif (t_1 <= 5e-201)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * sin(ky)) * sin(th));
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.72], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-201], N[(N[(N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.71999999999999997

   1. Initial program 87.2%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.3

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

   5. Applied rewrites 66.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. cos-2 N/A

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

      10. cos-sum N/A

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

      11. lower-cos.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-*.f64 54.0

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

   7. Applied rewrites 54.0%

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

   8. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. cos-neg-rev N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

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

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(2 \cdot ky\right)\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{-2} \cdot ky\right), \frac{1}{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 57.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(-2 \cdot ky\right)}, 0.5\right)}} \cdot \sin th \]

   10. Applied rewrites 57.5%

       \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}}} \cdot \sin th \]

   if -0.71999999999999997 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      11. lower-sin.f64 99.4

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 80.4%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \left(2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

       \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 49.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 49.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 61.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot ky\right) \cdot \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (* (/ (sin ky) (sqrt (fma -0.5 (cos (* -2.0 ky)) 0.5))) (sin th))
     (if (<= t_1 5e-201)
       (*
        (* (* (sqrt 2.0) ky) (sqrt (pow (- 1.0 (cos (* -2.0 kx))) -1.0)))
        (sin th))
       (if (<= t_1 0.02) (* (/ (sin ky) (sin kx)) (sin th)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (sin(ky) / sqrt(fma(-0.5, cos((-2.0 * ky)), 0.5))) * sin(th);
	} else if (t_1 <= 5e-201) {
		tmp = ((sqrt(2.0) * ky) * sqrt(pow((1.0 - cos((-2.0 * kx))), -1.0))) * sin(th);
	} else if (t_1 <= 0.02) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(-0.5, cos(Float64(-2.0 * ky)), 0.5))) * sin(th));
	elseif (t_1 <= 5e-201)
		tmp = Float64(Float64(Float64(sqrt(2.0) * ky) * sqrt((Float64(1.0 - cos(Float64(-2.0 * kx))) ^ -1.0))) * sin(th));
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-201], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 89.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 54.0

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 54.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      9. cos-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\left(\cos ky \cdot \cos ky - \sin ky \cdot \sin ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      10. cos-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 44.1

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   7. Applied rewrites 44.1%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}}} \cdot \sin th \]

      4. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{-1}{2}} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}}} \cdot \sin th \]

      5. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{-1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)} + \frac{1}{2}}} \cdot \sin th \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{-1}{2} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)} + \frac{1}{2}}} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{-1}{2} \cdot \cos \left(\color{blue}{-2} \cdot ky\right) + \frac{1}{2}}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(-2 \cdot ky\right), \frac{1}{2}\right)}}} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot ky\right), \frac{1}{2}\right)}} \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(2 \cdot ky\right)\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{-2} \cdot ky\right), \frac{1}{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 49.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \color{blue}{\left(-2 \cdot ky\right)}, 0.5\right)}} \cdot \sin th \]

   10. Applied rewrites 49.7%

       \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}}} \cdot \sin th \]

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      11. lower-sin.f64 99.5

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 75.2%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\left(ky \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 72.6%

       \[\leadsto \left(\left(\sqrt{2} \cdot ky\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}}}\right) \cdot \sin th \]

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 49.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 49.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot ky\right) \cdot \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{t\_1}^{-1}}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot ky\right) \cdot \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.1)
     (* (* (sin ky) th) (sqrt (pow t_1 -1.0)))
     (if (<= t_2 5e-201)
       (*
        (* (* (sqrt 2.0) ky) (sqrt (pow (- 1.0 (cos (* -2.0 kx))) -1.0)))
        (sin th))
       (if (<= t_2 0.02) (* (/ (sin ky) (sin kx)) (sin th)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = (sin(ky) * th) * sqrt(pow(t_1, -1.0));
	} else if (t_2 <= 5e-201) {
		tmp = ((sqrt(2.0) * ky) * sqrt(pow((1.0 - cos((-2.0 * kx))), -1.0))) * sin(th);
	} else if (t_2 <= 0.02) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = 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) :: t_2
    real(8) :: tmp
    t_1 = sin(ky) ** 2.0d0
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + t_1))
    if (t_2 <= (-0.1d0)) then
        tmp = (sin(ky) * th) * sqrt((t_1 ** (-1.0d0)))
    else if (t_2 <= 5d-201) then
        tmp = ((sqrt(2.0d0) * ky) * sqrt(((1.0d0 - cos(((-2.0d0) * kx))) ** (-1.0d0)))) * sin(th)
    else if (t_2 <= 0.02d0) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = (Math.sin(ky) * th) * Math.sqrt(Math.pow(t_1, -1.0));
	} else if (t_2 <= 5e-201) {
		tmp = ((Math.sqrt(2.0) * ky) * Math.sqrt(Math.pow((1.0 - Math.cos((-2.0 * kx))), -1.0))) * Math.sin(th);
	} else if (t_2 <= 0.02) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	tmp = 0
	if t_2 <= -0.1:
		tmp = (math.sin(ky) * th) * math.sqrt(math.pow(t_1, -1.0))
	elif t_2 <= 5e-201:
		tmp = ((math.sqrt(2.0) * ky) * math.sqrt(math.pow((1.0 - math.cos((-2.0 * kx))), -1.0))) * math.sin(th)
	elif t_2 <= 0.02:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = Float64(Float64(sin(ky) * th) * sqrt((t_1 ^ -1.0)));
	elseif (t_2 <= 5e-201)
		tmp = Float64(Float64(Float64(sqrt(2.0) * ky) * sqrt((Float64(1.0 - cos(Float64(-2.0 * kx))) ^ -1.0))) * sin(th));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	tmp = 0.0;
	if (t_2 <= -0.1)
		tmp = (sin(ky) * th) * sqrt((t_1 ^ -1.0));
	elseif (t_2 <= 5e-201)
		tmp = ((sqrt(2.0) * ky) * sqrt(((1.0 - cos((-2.0 * kx))) ^ -1.0))) * sin(th);
	elseif (t_2 <= 0.02)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[Power[t$95$1, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-201], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 89.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \]

      9. lower-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \]

      12. lower-sin.f64 51.4

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \]

   6. Taylor expanded in kx around 0

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{{\sin ky}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.5%

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{{\sin ky}^{2}}} \]

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      11. lower-sin.f64 99.5

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 75.2%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\left(ky \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 72.6%

       \[\leadsto \left(\left(\sqrt{2} \cdot ky\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}}}\right) \cdot \sin th \]

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 49.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 49.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \sqrt{{\left({\sin ky}^{2}\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot ky\right) \cdot \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot ky\right) \cdot \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (* (fma (* th th) -0.16666666666666666 1.0) th))
     (if (<= t_1 5e-201)
       (*
        (* (* (sqrt 2.0) ky) (sqrt (pow (- 1.0 (cos (* -2.0 kx))) -1.0)))
        (sin th))
       (if (<= t_1 0.02) (* (/ ky (sin kx)) (sin th)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_1 <= 5e-201) {
		tmp = ((sqrt(2.0) * ky) * sqrt(pow((1.0 - cos((-2.0 * kx))), -1.0))) * sin(th);
	} else if (t_1 <= 0.02) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_1 <= 5e-201)
		tmp = Float64(Float64(Float64(sqrt(2.0) * ky) * sqrt((Float64(1.0 - cos(Float64(-2.0 * kx))) ^ -1.0))) * sin(th));
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-201], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 89.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 54.0

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 54.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      9. cos-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\left(\cos ky \cdot \cos ky - \sin ky \cdot \sin ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      10. cos-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 44.1

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   7. Applied rewrites 44.1%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th\right) \]

      7. lower-*.f64 28.7

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th\right) \]

   10. Applied rewrites 28.7%

       \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      11. lower-sin.f64 99.5

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 75.2%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\left(ky \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 72.6%

       \[\leadsto \left(\left(\sqrt{2} \cdot ky\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}}}\right) \cdot \sin th \]

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 49.1

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot ky\right) \cdot \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -1.0)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (* (fma (* th th) -0.16666666666666666 1.0) th))
     (if (<= t_1 0.02) (* (/ (sin ky) (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_1 <= 0.02) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 83.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 83.0

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 83.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      9. cos-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\left(\cos ky \cdot \cos ky - \sin ky \cdot \sin ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      10. cos-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 66.5

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   7. Applied rewrites 66.5%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th\right) \]

      7. lower-*.f64 43.7

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th\right) \]

   10. Applied rewrites 43.7%

       \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 49.3

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 49.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 51.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -1.0)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (* (fma (* th th) -0.16666666666666666 1.0) th))
     (if (<= t_1 0.02) (* (/ (sin th) (sin kx)) (sin ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_1 <= 0.02) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 83.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 83.0

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 83.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      9. cos-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\left(\cos ky \cdot \cos ky - \sin ky \cdot \sin ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      10. cos-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 66.5

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   7. Applied rewrites 66.5%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th\right) \]

      7. lower-*.f64 43.7

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th\right) \]

   10. Applied rewrites 43.7%

       \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.5

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.4

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th}}{\sin kx} \cdot \sin ky \]

      3. lower-sin.f64 49.4

         \[\leadsto \frac{\sin th}{\color{blue}{\sin kx}} \cdot \sin ky \]

   7. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 47.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot ky\right) \cdot \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 5e-201)
     (*
      (* (* (sqrt 2.0) ky) (sqrt (pow (- 1.0 (cos (* -2.0 kx))) -1.0)))
      (sin th))
     (if (<= t_1 0.02) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-201) {
		tmp = ((sqrt(2.0) * ky) * sqrt(pow((1.0 - cos((-2.0 * kx))), -1.0))) * sin(th);
	} else if (t_1 <= 0.02) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = 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-201) then
        tmp = ((sqrt(2.0d0) * ky) * sqrt(((1.0d0 - cos(((-2.0d0) * kx))) ** (-1.0d0)))) * sin(th)
    else if (t_1 <= 0.02d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-201) {
		tmp = ((Math.sqrt(2.0) * ky) * Math.sqrt(Math.pow((1.0 - Math.cos((-2.0 * kx))), -1.0))) * Math.sin(th);
	} else if (t_1 <= 0.02) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 5e-201:
		tmp = ((math.sqrt(2.0) * ky) * math.sqrt(math.pow((1.0 - math.cos((-2.0 * kx))), -1.0))) * math.sin(th)
	elif t_1 <= 0.02:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 5e-201)
		tmp = Float64(Float64(Float64(sqrt(2.0) * ky) * sqrt((Float64(1.0 - cos(Float64(-2.0 * kx))) ^ -1.0))) * sin(th));
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 5e-201)
		tmp = ((sqrt(2.0) * ky) * sqrt(((1.0 - cos((-2.0 * kx))) ^ -1.0))) * sin(th);
	elseif (t_1 <= 0.02)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-201], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999999e-201

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      11. lower-sin.f64 93.0

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   5. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 77.8%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\left(ky \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}}\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 31.8%

       \[\leadsto \left(\left(\sqrt{2} \cdot ky\right) \cdot \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}}}\right) \cdot \sin th \]

   if 4.9999999999999999e-201 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 49.1

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-201}:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot ky\right) \cdot \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 45.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.02)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.02) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = 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(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.02d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.02) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.02:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 35.5

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 43.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.02)
   (*
    (/
     (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
     (sqrt (+ (* kx kx) (* ky ky))))
    (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.02) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(Float64(Float64(kx * kx) + Float64(ky * ky)))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 48.9

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 48.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

      2. lower-*.f64 29.5

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   8. Applied rewrites 29.5%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th \]

       3. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th \]

       4. *-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th \]

       6. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th \]

       7. lower-*.f64 28.0

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th \]

   11. Applied rewrites 28.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 15.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 2 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      2e-311)
   (* (* (* -0.16666666666666666 th) th) th)
   (*
    (fma
     (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
     (* th th)
     1.0)
    th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 2e-311) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-311)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 2e-311], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 1.9999999999999e-311

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 21.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 21.2%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.9%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 13.6%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 13.6%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 1.9999999999999e-311 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 31.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.8%

      \[\leadsto \mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{2}{2 - \left(\cos \left(-2 \cdot kx\right) + \cos \left(-2 \cdot ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2.5e-14)
   (*
    (/
     (sin ky)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin th))
   (*
    (* (sqrt (/ 2.0 (- 2.0 (+ (cos (* -2.0 kx)) (cos (* -2.0 ky)))))) (sin ky))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 2.5e-14) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = (sqrt((2.0 / (2.0 - (cos((-2.0 * kx)) + cos((-2.0 * ky)))))) * sin(ky)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 2.5e-14)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(2.0 - Float64(cos(Float64(-2.0 * kx)) + cos(Float64(-2.0 * ky)))))) * sin(ky)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 2.5e-14], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 / N[(2.0 - N[(N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.5000000000000001e-14

   1. Initial program 84.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.9

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if 2.5000000000000001e-14 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      11. lower-sin.f64 99.4

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 99.0%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in kx around inf

      \[\leadsto \left(\sqrt{\frac{2}{2 - \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 99.0%

       \[\leadsto \left(\sqrt{\frac{2}{\left(2 - \cos \left(-2 \cdot ky\right)\right) - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]
   11. Step-by-step derivation

   12. Applied rewrites 98.9%

       \[\leadsto \left(\sqrt{\frac{2}{2 - \left(\cos \left(-2 \cdot kx\right) + \cos \left(-2 \cdot ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 15.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 2 \cdot 10^{-311}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      2e-311)
   (* (* (* -0.16666666666666666 th) th) th)
   (* (fma (* -0.16666666666666666 th) th 1.0) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 2e-311) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = fma((-0.16666666666666666 * th), th, 1.0) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-311)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = Float64(fma(Float64(-0.16666666666666666 * th), th, 1.0) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 2e-311], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th + 1.0), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 1.9999999999999e-311

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 21.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 21.2%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.9%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 13.6%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 13.6%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 1.9999999999999e-311 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 31.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.9%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]
   9. Step-by-step derivation

   10. Applied rewrites 14.9%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \cdot th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 36.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{th}{\sin kx} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.02)
   (* (/ th (sin kx)) ky)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.02) {
		tmp = (th / sin(kx)) * ky;
	} else {
		tmp = 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(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.02d0) then
        tmp = (th / sin(kx)) * ky
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.02) {
		tmp = (th / Math.sin(kx)) * ky;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.02:
		tmp = (th / math.sin(kx)) * ky
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = Float64(Float64(th / sin(kx)) * ky);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = (th / sin(kx)) * ky;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot th\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \]

      9. lower-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \]

      12. lower-sin.f64 44.5

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \]

   5. Applied rewrites 44.5%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.8%

      \[\leadsto \frac{th}{\sin kx} \cdot \color{blue}{ky} \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 63.8

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 31.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1.02 \cdot 10^{-42}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
      1.02e-42)
   (* (* (* -0.16666666666666666 th) th) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 1.02e-42) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = 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(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 1.02d-42) then
        tmp = (((-0.16666666666666666d0) * th) * th) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 1.02e-42) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 1.02e-42:
		tmp = ((-0.16666666666666666 * th) * th) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1.02e-42)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1.02e-42)
		tmp = ((-0.16666666666666666 * th) * th) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.02e-42], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.0199999999999999e-42

   1. Initial program 94.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.2%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 12.3%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 12.3%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 1.0199999999999999e-42 < (/.f64 (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 88.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 59.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 78.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.45 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{1 - \left(\cos \left(-2 \cdot kx\right) - \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}{2}\right)}^{-1}} \cdot \sin ky\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.45e-7)
   (*
    (/
     (sin ky)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin th))
   (*
    (*
     (sqrt
      (pow
       (/ (- 1.0 (- (cos (* -2.0 kx)) (- 1.0 (cos (* -2.0 ky))))) 2.0)
       -1.0))
     (sin ky))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.45e-7) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = (sqrt(pow(((1.0 - (cos((-2.0 * kx)) - (1.0 - cos((-2.0 * ky))))) / 2.0), -1.0)) * sin(ky)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.45e-7)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(sqrt((Float64(Float64(1.0 - Float64(cos(Float64(-2.0 * kx)) - Float64(1.0 - cos(Float64(-2.0 * ky))))) / 2.0) ^ -1.0)) * sin(ky)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.45e-7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[Power[N[(N[(1.0 - N[(N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.4499999999999999e-7

   1. Initial program 89.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.8

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower-*.f64 73.7

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 73.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if 1.4499999999999999e-7 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      11. lower-sin.f64 99.4

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 99.0%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 99.0%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{1 - \left(\cos \left(-2 \cdot kx\right) - \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 1.45 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(\frac{1 - \left(\cos \left(-2 \cdot kx\right) - \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}{2}\right)}^{-1}} \cdot \sin ky\right) \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.45 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.45e-7)
   (*
    (/
     (sin ky)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin th))
   (*
    (/
     (sin ky)
     (/
      (sqrt
       (fma (- 1.0 (cos (* 2.0 ky))) 2.0 (* 2.0 (- 1.0 (cos (* 2.0 kx))))))
      2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.45e-7) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, (2.0 * (1.0 - cos((2.0 * kx)))))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.45e-7)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 2.0, Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * kx)))))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.45e-7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.4499999999999999e-7

   1. Initial program 89.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.8

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower-*.f64 73.7

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 73.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if 1.4499999999999999e-7 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.4

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      8. frac-add N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      11. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{\color{blue}{4}}}} \cdot \sin th \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\color{blue}{2}}} \cdot \sin th \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{2}}} \cdot \sin th \]

   6. Applied rewrites 99.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.45 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{2}{\left(2 - \cos \left(-2 \cdot ky\right)\right) - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.45e-7)
   (*
    (/
     (sin ky)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin th))
   (*
    (* (sqrt (/ 2.0 (- (- 2.0 (cos (* -2.0 ky))) (cos (* -2.0 kx))))) (sin ky))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.45e-7) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = (sqrt((2.0 / ((2.0 - cos((-2.0 * ky))) - cos((-2.0 * kx))))) * sin(ky)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.45e-7)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(Float64(2.0 - cos(Float64(-2.0 * ky))) - cos(Float64(-2.0 * kx))))) * sin(ky)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.45e-7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 / N[(N[(2.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.4499999999999999e-7

   1. Initial program 89.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.8

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower-*.f64 73.7

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 73.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if 1.4499999999999999e-7 < kx

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      11. lower-sin.f64 99.4

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 99.0%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in kx around inf

      \[\leadsto \left(\sqrt{\frac{2}{2 - \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 99.0%

       \[\leadsto \left(\sqrt{\frac{2}{\left(2 - \cos \left(-2 \cdot ky\right)\right) - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 44.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.35 \cdot 10^{-185}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 0.07:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 2.35e-185)
   (sin th)
   (if (<= kx 0.07)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (fma -0.5 (cos (* -2.0 ky)) 0.5))))
      (sin th))
     (* (* (sqrt (/ 2.0 (- 1.0 (cos (* -2.0 kx))))) (sin ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.35e-185) {
		tmp = sin(th);
	} else if (kx <= 0.07) {
		tmp = (sin(ky) / sqrt(((kx * kx) + fma(-0.5, cos((-2.0 * ky)), 0.5)))) * sin(th);
	} else {
		tmp = (sqrt((2.0 / (1.0 - cos((-2.0 * kx))))) * sin(ky)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 2.35e-185)
		tmp = sin(th);
	elseif (kx <= 0.07)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + fma(-0.5, cos(Float64(-2.0 * ky)), 0.5)))) * sin(th));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * sin(ky)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 2.35e-185], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 0.07], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(-0.5 * N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 2.3500000000000001e-185

   1. Initial program 86.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 33.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 33.3%

      \[\leadsto \color{blue}{\sin th} \]

   if 2.3500000000000001e-185 < kx < 0.070000000000000007

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 99.7

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      9. cos-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\left(\cos ky \cdot \cos ky - \sin ky \cdot \sin ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      10. cos-sum N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 84.3

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   7. Applied rewrites 84.3%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   8. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}\right)}}} \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\color{blue}{\frac{-1}{2}} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}\right)}} \cdot \sin th \]

      4. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{-1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)} + \frac{1}{2}\right)}} \cdot \sin th \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{-1}{2} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)} + \frac{1}{2}\right)}} \cdot \sin th \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{-1}{2} \cdot \cos \left(\color{blue}{-2} \cdot ky\right) + \frac{1}{2}\right)}} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(-2 \cdot ky\right), \frac{1}{2}\right)}}} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot ky\right), \frac{1}{2}\right)}} \cdot \sin th \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\mathsf{neg}\left(2 \cdot ky\right)\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{-2} \cdot ky\right), \frac{1}{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 84.3

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(-0.5, \cos \color{blue}{\left(-2 \cdot ky\right)}, 0.5\right)}} \cdot \sin th \]

   10. Applied rewrites 84.3%

       \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}}} \cdot \sin th \]

   if 0.070000000000000007 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \sin ky\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \sin ky\right) \cdot \sin th \]

      11. lower-sin.f64 99.4

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \sin ky\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 99.0%

      \[\leadsto \left(\sqrt{\frac{1}{\frac{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}{2}}} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \left(2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 47.7%

       \[\leadsto \left(\sqrt{\frac{2}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \sin ky\right) \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 10.9% accurate, 39.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (* (* -0.16666666666666666 th) th) th))

C

double code(double kx, double ky, double th) {
	return ((-0.16666666666666666 * th) * th) * th;
}

Fortran

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 = (((-0.16666666666666666d0) * th) * th) * th
end function

Java

public static double code(double kx, double ky, double th) {
	return ((-0.16666666666666666 * th) * th) * th;
}

Python

def code(kx, ky, th):
	return ((-0.16666666666666666 * th) * th) * th

Julia

function code(kx, ky, th)
	return Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = ((-0.16666666666666666 * th) * th) * th;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 26.0

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 26.0%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 13.3%

   \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

9. Taylor expanded in th around inf

   \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
10. Step-by-step derivation

11. Applied rewrites 9.0%

    \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
12. Step-by-step derivation

13. Applied rewrites 9.0%

    \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024350 
(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)))