Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 11.6s

Alternatives: 26

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: 93.8% 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 95.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-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. Add Preprocessing

Alternative 2: 84.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_2 := \mathsf{hypot}\left(t\_1, \sin kx\right)\\ t_3 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_4 := {\sin ky}^{2}\\ t_5 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_4}}\\ \mathbf{if}\;t\_5 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_4}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq -0.01:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq 10^{-7}:\\ \;\;\;\;\frac{t\_1}{t\_2} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.9999595807338642:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{t\_2} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_2 (hypot t_1 (sin kx)))
        (t_3 (/ (* (sin ky) th) (hypot (sin ky) (sin kx))))
        (t_4 (pow (sin ky) 2.0))
        (t_5 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_4)))))
   (if (<= t_5 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_4))) (sin th))
     (if (<= t_5 -0.01)
       t_3
       (if (<= t_5 1e-7)
         (* (/ t_1 t_2) (sin th))
         (if (<= t_5 0.9999595807338642)
           t_3
           (if (<= t_5 2.0) (sin th) (* (/ (sin th) t_2) (sin ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_2 = hypot(t_1, sin(kx));
	double t_3 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double t_4 = pow(sin(ky), 2.0);
	double t_5 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_4));
	double tmp;
	if (t_5 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_4))) * sin(th);
	} else if (t_5 <= -0.01) {
		tmp = t_3;
	} else if (t_5 <= 1e-7) {
		tmp = (t_1 / t_2) * sin(th);
	} else if (t_5 <= 0.9999595807338642) {
		tmp = t_3;
	} else if (t_5 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) / t_2) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_2 = hypot(t_1, sin(kx))
	t_3 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	t_4 = sin(ky) ^ 2.0
	t_5 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_4)))
	tmp = 0.0
	if (t_5 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_4))) * sin(th));
	elseif (t_5 <= -0.01)
		tmp = t_3;
	elseif (t_5 <= 1e-7)
		tmp = Float64(Float64(t_1 / t_2) * sin(th));
	elseif (t_5 <= 0.9999595807338642)
		tmp = t_3;
	elseif (t_5 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) / t_2) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -0.01], t$95$3, If[LessEqual[t$95$5, 1e-7], N[(N[(t$95$1 / t$95$2), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.9999595807338642], t$95$3, If[LessEqual[t$95$5, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / t$95$2), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 88.4%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0100000000000000002 or 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99995958073386415

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 56.2

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

   7. Applied rewrites 56.2%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999995e-8

   1. Initial program 99.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-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 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if 0.99995958073386415 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 2.4

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 3: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_4 := \mathsf{hypot}\left(t\_3, \sin kx\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)}\\ \mathbf{elif}\;t\_2 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-7}:\\ \;\;\;\;\frac{t\_3}{t\_4} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.9999595807338642:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{t\_4} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin ky) (sin kx))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_4 (hypot t_3 (sin kx))))
   (if (<= t_2 -1.0)
     (/
      (* (sin th) (sin ky))
      (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
     (if (<= t_2 -0.01)
       t_1
       (if (<= t_2 1e-7)
         (* (/ t_3 t_4) (sin th))
         (if (<= t_2 0.9999595807338642)
           t_1
           (if (<= t_2 2.0) (sin th) (* (/ (sin th) t_4) (sin ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_4 = hypot(t_3, sin(kx));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (sin(th) * sin(ky)) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx));
	} else if (t_2 <= -0.01) {
		tmp = t_1;
	} else if (t_2 <= 1e-7) {
		tmp = (t_3 / t_4) * sin(th);
	} else if (t_2 <= 0.9999595807338642) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) / t_4) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_4 = hypot(t_3, sin(kx))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(sin(th) * sin(ky)) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx)));
	elseif (t_2 <= -0.01)
		tmp = t_1;
	elseif (t_2 <= 1e-7)
		tmp = Float64(Float64(t_3 / t_4) * sin(th));
	elseif (t_2 <= 0.9999595807338642)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) / t_4) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $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[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$3 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.01], t$95$1, If[LessEqual[t$95$2, 1e-7], N[(N[(t$95$3 / t$95$4), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999595807338642], t$95$1, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / t$95$4), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   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-*.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 88.0

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

      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 92.9

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 92.9

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

   7. Applied rewrites 92.9%

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

   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))))) < -0.0100000000000000002 or 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99995958073386415

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 56.2

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

   7. Applied rewrites 56.2%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999995e-8

   1. Initial program 99.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-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 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if 0.99995958073386415 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 2.4

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 4: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_3 := \mathsf{hypot}\left(t\_2, \sin kx\right)\\ t_4 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;t\_1 \leq -0.95:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{1 - \cos \left(-2 \cdot ky\right)}}{\sqrt{2}}} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq -0.01:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 10^{-7}:\\ \;\;\;\;\frac{t\_2}{t\_3} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.9999595807338642:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{t\_3} \cdot \sin ky\\ \end{array} \end{array} \]

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 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_3 (hypot t_2 (sin kx)))
        (t_4 (/ (* (sin ky) th) (hypot (sin ky) (sin kx)))))
   (if (<= t_1 -0.95)
     (* (/ (sin th) (/ (sqrt (- 1.0 (cos (* -2.0 ky)))) (sqrt 2.0))) (sin ky))
     (if (<= t_1 -0.01)
       t_4
       (if (<= t_1 1e-7)
         (* (/ t_2 t_3) (sin th))
         (if (<= t_1 0.9999595807338642)
           t_4
           (if (<= t_1 2.0) (sin th) (* (/ (sin th) t_3) (sin ky)))))))))

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 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_3 = hypot(t_2, sin(kx));
	double t_4 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double tmp;
	if (t_1 <= -0.95) {
		tmp = (sin(th) / (sqrt((1.0 - cos((-2.0 * ky)))) / sqrt(2.0))) * sin(ky);
	} else if (t_1 <= -0.01) {
		tmp = t_4;
	} else if (t_1 <= 1e-7) {
		tmp = (t_2 / t_3) * sin(th);
	} else if (t_1 <= 0.9999595807338642) {
		tmp = t_4;
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) / t_3) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_3 = hypot(t_2, sin(kx))
	t_4 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	tmp = 0.0
	if (t_1 <= -0.95)
		tmp = Float64(Float64(sin(th) / Float64(sqrt(Float64(1.0 - cos(Float64(-2.0 * ky)))) / sqrt(2.0))) * sin(ky));
	elseif (t_1 <= -0.01)
		tmp = t_4;
	elseif (t_1 <= 1e-7)
		tmp = Float64(Float64(t_2 / t_3) * sin(th));
	elseif (t_1 <= 0.9999595807338642)
		tmp = t_4;
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) / t_3) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.95], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.01], t$95$4, If[LessEqual[t$95$1, 1e-7], N[(N[(t$95$2 / t$95$3), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999595807338642], t$95$4, If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / t$95$3), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 89.7%

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

      1. lift-*.f64 N/A

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

      2. 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 89.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 64.9%

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

   7. Taylor expanded in kx around 0

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

      1. lower--.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lower-cos.f64 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 56.0

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

   9. Applied rewrites 56.0%

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

   if -0.94999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0100000000000000002 or 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99995958073386415

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.0

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

   7. Applied rewrites 58.0%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999995e-8

   1. Initial program 99.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-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 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if 0.99995958073386415 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 2.4

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 5: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_2 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_4 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{if}\;t\_3 \leq -0.95:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{1 - \cos \left(-2 \cdot ky\right)}}{\sqrt{2}}} \cdot \sin ky\\ \mathbf{elif}\;t\_3 \leq -0.01:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-7}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.9999595807338642:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_2 (/ (* (sin ky) th) (hypot (sin ky) (sin kx))))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_4 (* (/ t_1 (hypot t_1 (sin kx))) (sin th))))
   (if (<= t_3 -0.95)
     (* (/ (sin th) (/ (sqrt (- 1.0 (cos (* -2.0 ky)))) (sqrt 2.0))) (sin ky))
     (if (<= t_3 -0.01)
       t_2
       (if (<= t_3 1e-7)
         t_4
         (if (<= t_3 0.9999595807338642)
           t_2
           (if (<= t_3 2.0) (sin th) t_4)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_2 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_4 = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	double tmp;
	if (t_3 <= -0.95) {
		tmp = (sin(th) / (sqrt((1.0 - cos((-2.0 * ky)))) / sqrt(2.0))) * sin(ky);
	} else if (t_3 <= -0.01) {
		tmp = t_2;
	} else if (t_3 <= 1e-7) {
		tmp = t_4;
	} else if (t_3 <= 0.9999595807338642) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_2 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_4 = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th))
	tmp = 0.0
	if (t_3 <= -0.95)
		tmp = Float64(Float64(sin(th) / Float64(sqrt(Float64(1.0 - cos(Float64(-2.0 * ky)))) / sqrt(2.0))) * sin(ky));
	elseif (t_3 <= -0.01)
		tmp = t_2;
	elseif (t_3 <= 1e-7)
		tmp = t_4;
	elseif (t_3 <= 0.9999595807338642)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $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]}, Block[{t$95$4 = N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.95], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.01], t$95$2, If[LessEqual[t$95$3, 1e-7], t$95$4, If[LessEqual[t$95$3, 0.9999595807338642], t$95$2, If[LessEqual[t$95$3, 2.0], N[Sin[th], $MachinePrecision], t$95$4]]]]]]]]]

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))))) < -0.94999999999999996

   1. Initial program 89.7%

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

      1. lift-*.f64 N/A

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

      2. 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 89.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 64.9%

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

   7. Taylor expanded in kx around 0

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

      1. lower--.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lower-cos.f64 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 56.0

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

   9. Applied rewrites 56.0%

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

   if -0.94999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0100000000000000002 or 9.9999999999999995e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99995958073386415

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.0

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

   7. Applied rewrites 58.0%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999995e-8 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.5%

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

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

   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 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if 0.99995958073386415 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 6: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ t_4 := \left(\left(\sqrt{2} \cdot th\right) \cdot \sqrt{{\left(\left(2 - \cos \left(-2 \cdot kx\right)\right) - \cos \left(2 \cdot ky\right)\right)}^{-1}}\right) \cdot \sin ky\\ \mathbf{if}\;t\_2 \leq -0.95:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{1 - \cos \left(-2 \cdot ky\right)}}{\sqrt{2}}} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq -0.01:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0.005:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.9999595807338642:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (* (/ t_1 (hypot t_1 (sin kx))) (sin th)))
        (t_4
         (*
          (*
           (* (sqrt 2.0) th)
           (sqrt (pow (- (- 2.0 (cos (* -2.0 kx))) (cos (* 2.0 ky))) -1.0)))
          (sin ky))))
   (if (<= t_2 -0.95)
     (* (/ (sin th) (/ (sqrt (- 1.0 (cos (* -2.0 ky)))) (sqrt 2.0))) (sin ky))
     (if (<= t_2 -0.01)
       t_4
       (if (<= t_2 0.005)
         t_3
         (if (<= t_2 0.9999595807338642)
           t_4
           (if (<= t_2 2.0) (sin th) t_3)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	double t_4 = ((sqrt(2.0) * th) * sqrt(pow(((2.0 - cos((-2.0 * kx))) - cos((2.0 * ky))), -1.0))) * sin(ky);
	double tmp;
	if (t_2 <= -0.95) {
		tmp = (sin(th) / (sqrt((1.0 - cos((-2.0 * ky)))) / sqrt(2.0))) * sin(ky);
	} else if (t_2 <= -0.01) {
		tmp = t_4;
	} else if (t_2 <= 0.005) {
		tmp = t_3;
	} else if (t_2 <= 0.9999595807338642) {
		tmp = t_4;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th))
	t_4 = Float64(Float64(Float64(sqrt(2.0) * th) * sqrt((Float64(Float64(2.0 - cos(Float64(-2.0 * kx))) - cos(Float64(2.0 * ky))) ^ -1.0))) * sin(ky))
	tmp = 0.0
	if (t_2 <= -0.95)
		tmp = Float64(Float64(sin(th) / Float64(sqrt(Float64(1.0 - cos(Float64(-2.0 * ky)))) / sqrt(2.0))) * sin(ky));
	elseif (t_2 <= -0.01)
		tmp = t_4;
	elseif (t_2 <= 0.005)
		tmp = t_3;
	elseif (t_2 <= 0.9999595807338642)
		tmp = t_4;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $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[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(2.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.95], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.01], t$95$4, If[LessEqual[t$95$2, 0.005], t$95$3, If[LessEqual[t$95$2, 0.9999595807338642], t$95$4, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], t$95$3]]]]]]]]]

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))))) < -0.94999999999999996

   1. Initial program 89.7%

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

      1. lift-*.f64 N/A

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

      2. 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 89.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 64.9%

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

   7. Taylor expanded in kx around 0

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

      1. lower--.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lower-cos.f64 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 56.0

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

   9. Applied rewrites 56.0%

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

   if -0.94999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0100000000000000002 or 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99995958073386415

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      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 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 97.5%

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

   7. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lower-cos.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sin.f64 20.5

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

   9. Applied rewrites 20.5%

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

   10. Taylor expanded in th around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. associate--r+ N/A

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

       8. lower--.f64 N/A

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

       9. lower--.f64 N/A

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

       10. cos-neg-rev N/A

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

       11. lower-cos.f64 N/A

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

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

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

       13. metadata-eval N/A

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

       14. lower-*.f64 N/A

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

       15. metadata-eval N/A

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

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

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

       17. lower-cos.f64 N/A

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

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

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

       19. metadata-eval N/A

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

       20. lower-*.f64 55.6

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

   12. Applied rewrites 55.6%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if 0.99995958073386415 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.95:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{1 - \cos \left(-2 \cdot ky\right)}}{\sqrt{2}}} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.01:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot th\right) \cdot \sqrt{{\left(\left(2 - \cos \left(-2 \cdot kx\right)\right) - \cos \left(2 \cdot ky\right)\right)}^{-1}}\right) \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9999595807338642:\\ \;\;\;\;\left(\left(\sqrt{2} \cdot th\right) \cdot \sqrt{{\left(\left(2 - \cos \left(-2 \cdot kx\right)\right) - \cos \left(2 \cdot ky\right)\right)}^{-1}}\right) \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \cos \left(-2 \cdot ky\right)\\ t_3 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_4 := \frac{t\_3}{\mathsf{hypot}\left(t\_3, \sin kx\right)} \cdot \sin th\\ t_5 := \left(\left(\sin ky \cdot th\right) \cdot \sqrt{2}\right) \cdot \sqrt{{\left(\left(2 - t\_2\right) - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\\ \mathbf{if}\;t\_1 \leq -0.95:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{1 - t\_2}}{\sqrt{2}}} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq -0.01:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 0.9999595807338642:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

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 ky)))
        (t_3 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_4 (* (/ t_3 (hypot t_3 (sin kx))) (sin th)))
        (t_5
         (*
          (* (* (sin ky) th) (sqrt 2.0))
          (sqrt (pow (- (- 2.0 t_2) (cos (* -2.0 kx))) -1.0)))))
   (if (<= t_1 -0.95)
     (* (/ (sin th) (/ (sqrt (- 1.0 t_2)) (sqrt 2.0))) (sin ky))
     (if (<= t_1 -0.01)
       t_5
       (if (<= t_1 0.005)
         t_4
         (if (<= t_1 0.9999595807338642)
           t_5
           (if (<= t_1 2.0) (sin th) t_4)))))))

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 * ky));
	double t_3 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_4 = (t_3 / hypot(t_3, sin(kx))) * sin(th);
	double t_5 = ((sin(ky) * th) * sqrt(2.0)) * sqrt(pow(((2.0 - t_2) - cos((-2.0 * kx))), -1.0));
	double tmp;
	if (t_1 <= -0.95) {
		tmp = (sin(th) / (sqrt((1.0 - t_2)) / sqrt(2.0))) * sin(ky);
	} else if (t_1 <= -0.01) {
		tmp = t_5;
	} else if (t_1 <= 0.005) {
		tmp = t_4;
	} else if (t_1 <= 0.9999595807338642) {
		tmp = t_5;
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = t_4;
	}
	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 * ky))
	t_3 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_4 = Float64(Float64(t_3 / hypot(t_3, sin(kx))) * sin(th))
	t_5 = Float64(Float64(Float64(sin(ky) * th) * sqrt(2.0)) * sqrt((Float64(Float64(2.0 - t_2) - cos(Float64(-2.0 * kx))) ^ -1.0)))
	tmp = 0.0
	if (t_1 <= -0.95)
		tmp = Float64(Float64(sin(th) / Float64(sqrt(Float64(1.0 - t_2)) / sqrt(2.0))) * sin(ky));
	elseif (t_1 <= -0.01)
		tmp = t_5;
	elseif (t_1 <= 0.005)
		tmp = t_4;
	elseif (t_1 <= 0.9999595807338642)
		tmp = t_5;
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = t_4;
	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 * ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 / N[Sqrt[t$95$3 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(2.0 - t$95$2), $MachinePrecision] - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.95], N[(N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.01], t$95$5, If[LessEqual[t$95$1, 0.005], t$95$4, If[LessEqual[t$95$1, 0.9999595807338642], t$95$5, If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], t$95$4]]]]]]]]]]

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))))) < -0.94999999999999996

   1. Initial program 89.7%

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

      1. lift-*.f64 N/A

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

      2. 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 89.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 64.9%

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

   7. Taylor expanded in kx around 0

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

      1. lower--.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. lower-cos.f64 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 56.0

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

   9. Applied rewrites 56.0%

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

   if -0.94999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0100000000000000002 or 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99995958073386415

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      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 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 97.5%

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

   7. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 N/A

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

      14. cos-neg-rev N/A

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

      15. lower-cos.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. cos-neg-rev N/A

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

      20. lower-cos.f64 N/A

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

   9. Applied rewrites 55.6%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if 0.99995958073386415 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.95:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{1 - \cos \left(-2 \cdot ky\right)}}{\sqrt{2}}} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.01:\\ \;\;\;\;\left(\left(\sin ky \cdot th\right) \cdot \sqrt{2}\right) \cdot \sqrt{{\left(\left(2 - \cos \left(-2 \cdot ky\right)\right) - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9999595807338642:\\ \;\;\;\;\left(\left(\sin ky \cdot th\right) \cdot \sqrt{2}\right) \cdot \sqrt{{\left(\left(2 - \cos \left(-2 \cdot ky\right)\right) - \cos \left(-2 \cdot kx\right)\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{if}\;t\_1 \leq -0.01:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\right)\right) \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-224}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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
         (*
          (sqrt (pow (- 1.0 (cos (* -2.0 kx))) -1.0))
          (* (* (sin th) ky) (sqrt 2.0)))))
   (if (<= t_1 -0.01)
     (*
      (*
       (sqrt (pow (- 1.0 (cos (* -2.0 ky))) -1.0))
       (*
        (sqrt 2.0)
        (*
         (fma
          (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
          (* th th)
          1.0)
         th)))
      (sin ky))
     (if (<= t_1 2e-224)
       t_2
       (if (<= t_1 2e-118)
         (* (/ ky (sin kx)) (sin th))
         (if (<= t_1 0.005) t_2 (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 = sqrt(pow((1.0 - cos((-2.0 * kx))), -1.0)) * ((sin(th) * ky) * sqrt(2.0));
	double tmp;
	if (t_1 <= -0.01) {
		tmp = (sqrt(pow((1.0 - cos((-2.0 * ky))), -1.0)) * (sqrt(2.0) * (fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th))) * sin(ky);
	} else if (t_1 <= 2e-224) {
		tmp = t_2;
	} else if (t_1 <= 2e-118) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_1 <= 0.005) {
		tmp = t_2;
	} 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 = Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * kx))) ^ -1.0)) * Float64(Float64(sin(th) * ky) * sqrt(2.0)))
	tmp = 0.0
	if (t_1 <= -0.01)
		tmp = Float64(Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * ky))) ^ -1.0)) * Float64(sqrt(2.0) * Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th))) * sin(ky));
	elseif (t_1 <= 2e-224)
		tmp = t_2;
	elseif (t_1 <= 2e-118)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_1 <= 0.005)
		tmp = t_2;
	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[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.01], N[(N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-224], t$95$2, If[LessEqual[t$95$1, 2e-118], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.005], t$95$2, 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))))) < -0.0100000000000000002

   1. Initial program 93.5%

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

      1. lift-*.f64 N/A

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

      2. 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 93.5

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 78.3%

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

   7. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lower-cos.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sin.f64 41.7

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

   9. Applied rewrites 41.7%

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

   10. Taylor expanded in th around 0

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

   12. Applied rewrites 20.0%

       \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot \color{blue}{th}\right)\right)\right) \cdot \sin ky \]

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-224 or 1.99999999999999997e-118 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 99.0%

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

      1. lift-*.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 98.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 75.6%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lower-cos.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-sqrt.f64 74.6

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

   9. Applied rewrites 74.6%

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

   if 2e-224 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.99999999999999997e-118

   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 55.7

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

   5. Applied rewrites 55.7%

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

   if 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.6

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

   5. Applied rewrites 67.6%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.01:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\right)\right) \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.01:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot th\right)\right) \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.005:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (sqrt (pow (- 1.0 (cos (* -2.0 kx))) -1.0))
          (* (* (sin th) ky) (sqrt 2.0))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.01)
     (*
      (* (sqrt (pow (- 1.0 (cos (* -2.0 ky))) -1.0)) (* (sqrt 2.0) th))
      (sin ky))
     (if (<= t_2 2e-224)
       t_1
       (if (<= t_2 2e-118)
         (* (/ ky (sin kx)) (sin th))
         (if (<= t_2 0.005) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sqrt(pow((1.0 - cos((-2.0 * kx))), -1.0)) * ((sin(th) * ky) * sqrt(2.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.01) {
		tmp = (sqrt(pow((1.0 - cos((-2.0 * ky))), -1.0)) * (sqrt(2.0) * th)) * sin(ky);
	} else if (t_2 <= 2e-224) {
		tmp = t_1;
	} else if (t_2 <= 2e-118) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_2 <= 0.005) {
		tmp = t_1;
	} 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 = sqrt(((1.0d0 - cos(((-2.0d0) * kx))) ** (-1.0d0))) * ((sin(th) * ky) * sqrt(2.0d0))
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_2 <= (-0.01d0)) then
        tmp = (sqrt(((1.0d0 - cos(((-2.0d0) * ky))) ** (-1.0d0))) * (sqrt(2.0d0) * th)) * sin(ky)
    else if (t_2 <= 2d-224) then
        tmp = t_1
    else if (t_2 <= 2d-118) then
        tmp = (ky / sin(kx)) * sin(th)
    else if (t_2 <= 0.005d0) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sqrt(Math.pow((1.0 - Math.cos((-2.0 * kx))), -1.0)) * ((Math.sin(th) * ky) * Math.sqrt(2.0));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.01) {
		tmp = (Math.sqrt(Math.pow((1.0 - Math.cos((-2.0 * ky))), -1.0)) * (Math.sqrt(2.0) * th)) * Math.sin(ky);
	} else if (t_2 <= 2e-224) {
		tmp = t_1;
	} else if (t_2 <= 2e-118) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else if (t_2 <= 0.005) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sqrt(math.pow((1.0 - math.cos((-2.0 * kx))), -1.0)) * ((math.sin(th) * ky) * math.sqrt(2.0))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.01:
		tmp = (math.sqrt(math.pow((1.0 - math.cos((-2.0 * ky))), -1.0)) * (math.sqrt(2.0) * th)) * math.sin(ky)
	elif t_2 <= 2e-224:
		tmp = t_1
	elif t_2 <= 2e-118:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	elif t_2 <= 0.005:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * kx))) ^ -1.0)) * Float64(Float64(sin(th) * ky) * sqrt(2.0)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.01)
		tmp = Float64(Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * ky))) ^ -1.0)) * Float64(sqrt(2.0) * th)) * sin(ky));
	elseif (t_2 <= 2e-224)
		tmp = t_1;
	elseif (t_2 <= 2e-118)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_2 <= 0.005)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sqrt(((1.0 - cos((-2.0 * kx))) ^ -1.0)) * ((sin(th) * ky) * sqrt(2.0));
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.01)
		tmp = (sqrt(((1.0 - cos((-2.0 * ky))) ^ -1.0)) * (sqrt(2.0) * th)) * sin(ky);
	elseif (t_2 <= 2e-224)
		tmp = t_1;
	elseif (t_2 <= 2e-118)
		tmp = (ky / sin(kx)) * sin(th);
	elseif (t_2 <= 0.005)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[2.0], $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, -0.01], N[(N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-224], t$95$1, If[LessEqual[t$95$2, 2e-118], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.005], t$95$1, 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))))) < -0.0100000000000000002

   1. Initial program 93.5%

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

      1. lift-*.f64 N/A

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

      2. 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 93.5

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 78.3%

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

   7. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lower-cos.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sin.f64 41.7

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

   9. Applied rewrites 41.7%

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

   10. Taylor expanded in th around 0

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

   12. Applied rewrites 19.7%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-224 or 1.99999999999999997e-118 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 99.0%

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

      1. lift-*.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 98.9

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 75.6%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lower-cos.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-sqrt.f64 74.6

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

   9. Applied rewrites 74.6%

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

   if 2e-224 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.99999999999999997e-118

   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 55.7

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

   5. Applied rewrites 55.7%

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

   if 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.6

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

   5. Applied rewrites 67.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.01:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot th\right)\right) \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.005:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (sqrt (pow (- 1.0 (cos (* -2.0 kx))) -1.0))
          (* (* (sin th) ky) (sqrt 2.0))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 2e-224)
     t_1
     (if (<= t_2 2e-118)
       (* (/ ky (sin kx)) (sin th))
       (if (<= t_2 0.005) t_1 (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sqrt(pow((1.0 - cos((-2.0 * kx))), -1.0)) * ((sin(th) * ky) * sqrt(2.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 2e-224) {
		tmp = t_1;
	} else if (t_2 <= 2e-118) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_2 <= 0.005) {
		tmp = t_1;
	} 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 = sqrt(((1.0d0 - cos(((-2.0d0) * kx))) ** (-1.0d0))) * ((sin(th) * ky) * sqrt(2.0d0))
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_2 <= 2d-224) then
        tmp = t_1
    else if (t_2 <= 2d-118) then
        tmp = (ky / sin(kx)) * sin(th)
    else if (t_2 <= 0.005d0) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sqrt(Math.pow((1.0 - Math.cos((-2.0 * kx))), -1.0)) * ((Math.sin(th) * ky) * Math.sqrt(2.0));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 2e-224) {
		tmp = t_1;
	} else if (t_2 <= 2e-118) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else if (t_2 <= 0.005) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sqrt(math.pow((1.0 - math.cos((-2.0 * kx))), -1.0)) * ((math.sin(th) * ky) * math.sqrt(2.0))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= 2e-224:
		tmp = t_1
	elif t_2 <= 2e-118:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	elif t_2 <= 0.005:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * kx))) ^ -1.0)) * Float64(Float64(sin(th) * ky) * sqrt(2.0)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 2e-224)
		tmp = t_1;
	elseif (t_2 <= 2e-118)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_2 <= 0.005)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sqrt(((1.0 - cos((-2.0 * kx))) ^ -1.0)) * ((sin(th) * ky) * sqrt(2.0));
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= 2e-224)
		tmp = t_1;
	elseif (t_2 <= 2e-118)
		tmp = (ky / sin(kx)) * sin(th);
	elseif (t_2 <= 0.005)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[2.0], $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, 2e-224], t$95$1, If[LessEqual[t$95$2, 2e-118], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.005], t$95$1, 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))))) < 2e-224 or 1.99999999999999997e-118 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 96.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 96.1

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 77.0%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lower-cos.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-sqrt.f64 36.4

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

   9. Applied rewrites 36.4%

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

   if 2e-224 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.99999999999999997e-118

   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 55.7

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

   5. Applied rewrites 55.7%

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

   if 0.0050000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 67.6

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

   5. Applied rewrites 67.6%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.005:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.01:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\right)\right) \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.01)
     (*
      (*
       (sqrt (pow (- 1.0 (cos (* -2.0 ky))) -1.0))
       (*
        (sqrt 2.0)
        (*
         (fma
          (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
          (* th th)
          1.0)
         th)))
      (sin ky))
     (if (<= t_1 2e-224)
       (*
        (sqrt (pow (- 1.0 (cos (* -2.0 kx))) -1.0))
        (* (* (sin th) ky) (sqrt 2.0)))
       (if (<= t_1 0.1) (* (/ (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 <= -0.01) {
		tmp = (sqrt(pow((1.0 - cos((-2.0 * ky))), -1.0)) * (sqrt(2.0) * (fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th))) * sin(ky);
	} else if (t_1 <= 2e-224) {
		tmp = sqrt(pow((1.0 - cos((-2.0 * kx))), -1.0)) * ((sin(th) * ky) * sqrt(2.0));
	} else if (t_1 <= 0.1) {
		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 <= -0.01)
		tmp = Float64(Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * ky))) ^ -1.0)) * Float64(sqrt(2.0) * Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th))) * sin(ky));
	elseif (t_1 <= 2e-224)
		tmp = Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * kx))) ^ -1.0)) * Float64(Float64(sin(th) * ky) * sqrt(2.0)));
	elseif (t_1 <= 0.1)
		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, -0.01], N[(N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-224], N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], 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))))) < -0.0100000000000000002

   1. Initial program 93.5%

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

      1. lift-*.f64 N/A

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

      2. 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 93.5

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 78.3%

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

   7. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lower-cos.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sin.f64 41.7

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

   9. Applied rewrites 41.7%

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

   10. Taylor expanded in th around 0

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

   12. Applied rewrites 20.0%

       \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot \color{blue}{th}\right)\right)\right) \cdot \sin ky \]

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-224

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. 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 98.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 74.2%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. cos-neg-rev N/A

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

      7. lower-cos.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-sqrt.f64 73.1

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

   9. Applied rewrites 73.1%

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

   if 2e-224 < (/.f64 (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 99.4%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 62.1

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

   5. Applied rewrites 62.1%

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

   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)))))

   1. Initial program 93.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 68.7

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

   5. Applied rewrites 68.7%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.01:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\right)\right) \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.01:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\right)\right) \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.01)
     (*
      (*
       (sqrt (pow (- 1.0 (cos (* -2.0 ky))) -1.0))
       (*
        (sqrt 2.0)
        (*
         (fma
          (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
          (* th th)
          1.0)
         th)))
      (sin ky))
     (if (<= t_1 2e-224)
       (*
        (sqrt (pow (- 1.0 (cos (* -2.0 kx))) -1.0))
        (* (* (sin th) ky) (sqrt 2.0)))
       (if (<= t_1 0.1) (* (/ (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 <= -0.01) {
		tmp = (sqrt(pow((1.0 - cos((-2.0 * ky))), -1.0)) * (sqrt(2.0) * (fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th))) * sin(ky);
	} else if (t_1 <= 2e-224) {
		tmp = sqrt(pow((1.0 - cos((-2.0 * kx))), -1.0)) * ((sin(th) * ky) * sqrt(2.0));
	} else if (t_1 <= 0.1) {
		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 <= -0.01)
		tmp = Float64(Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * ky))) ^ -1.0)) * Float64(sqrt(2.0) * Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th))) * sin(ky));
	elseif (t_1 <= 2e-224)
		tmp = Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * kx))) ^ -1.0)) * Float64(Float64(sin(th) * ky) * sqrt(2.0)));
	elseif (t_1 <= 0.1)
		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, -0.01], N[(N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-224], N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], 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 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))))) < -0.0100000000000000002

   1. Initial program 93.5%

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

      1. lift-*.f64 N/A

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

      2. 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 93.5

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin ky \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin ky \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin ky \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin ky \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   6. Applied rewrites 78.3%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   7. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\left(\left(\sin th \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}\right)} \cdot \sin ky \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \cdot \sin ky \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \cdot \sin ky \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      6. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      10. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{-2} \cdot ky\right)}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \color{blue}{\left(\sqrt{2} \cdot \sin th\right)}\right) \cdot \sin ky \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \color{blue}{\left(\sqrt{2} \cdot \sin th\right)}\right) \cdot \sin ky \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\color{blue}{\sqrt{2}} \cdot \sin th\right)\right) \cdot \sin ky \]

      14. lower-sin.f64 41.7

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \color{blue}{\sin th}\right)\right) \cdot \sin ky \]

   9. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \sin th\right)\right)} \cdot \sin ky \]

   10. Taylor expanded in th around 0

       \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \left(th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \cdot \sin ky \]
   11. Step-by-step derivation

   12. Applied rewrites 20.0%

       \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot \color{blue}{th}\right)\right)\right) \cdot \sin ky \]

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-224

   1. Initial program 98.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. 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 98.7

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.4

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin ky \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin ky \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin ky \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin ky \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin ky \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   6. Applied rewrites 74.2%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      6. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(\color{blue}{-2} \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{2}\right) \]

      15. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\color{blue}{\sin th} \cdot ky\right) \cdot \sqrt{2}\right) \]

      16. lower-sqrt.f64 73.1

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \color{blue}{\sqrt{2}}\right) \]

   9. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)} \]

   if 2e-224 < (/.f64 (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 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.7

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.7

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   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 62.3

         \[\leadsto \frac{\sin th}{\color{blue}{\sin kx}} \cdot \sin ky \]

   7. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]

   if 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 68.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.01:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\right)\right) \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot kx\right)\right)}^{-1}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\right)\right) \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.708:\\ \;\;\;\;\frac{\sin th}{\left|\sin kx\right|} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.99)
     (*
      (*
       (sqrt (pow (- 1.0 (cos (* -2.0 ky))) -1.0))
       (*
        (sqrt 2.0)
        (*
         (fma
          (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
          (* th th)
          1.0)
         th)))
      (sin ky))
     (if (<= t_1 0.708) (* (/ (sin th) (fabs (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 <= -0.99) {
		tmp = (sqrt(pow((1.0 - cos((-2.0 * ky))), -1.0)) * (sqrt(2.0) * (fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th))) * sin(ky);
	} else if (t_1 <= 0.708) {
		tmp = (sin(th) / fabs(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 <= -0.99)
		tmp = Float64(Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * ky))) ^ -1.0)) * Float64(sqrt(2.0) * Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th))) * sin(ky));
	elseif (t_1 <= 0.708)
		tmp = Float64(Float64(sin(th) / abs(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, -0.99], N[(N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.708], N[(N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999

   1. Initial program 88.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. 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 88.8

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.8

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin ky \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin ky \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin ky \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin ky \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin ky \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   6. Applied rewrites 61.9%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   7. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\left(\left(\sin th \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}\right)} \cdot \sin ky \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \cdot \sin ky \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \cdot \sin ky \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      6. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      10. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{-2} \cdot ky\right)}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \color{blue}{\left(\sqrt{2} \cdot \sin th\right)}\right) \cdot \sin ky \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \color{blue}{\left(\sqrt{2} \cdot \sin th\right)}\right) \cdot \sin ky \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\color{blue}{\sqrt{2}} \cdot \sin th\right)\right) \cdot \sin ky \]

      14. lower-sin.f64 58.7

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \color{blue}{\sin th}\right)\right) \cdot \sin ky \]

   9. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \sin th\right)\right)} \cdot \sin ky \]

   10. Taylor expanded in th around 0

       \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \left(th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \cdot \sin ky \]
   11. Step-by-step derivation

   12. Applied rewrites 27.3%

       \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot \color{blue}{th}\right)\right)\right) \cdot \sin ky \]

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.70799999999999996

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. 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 99.2

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.5

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   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 45.1

         \[\leadsto \frac{\sin th}{\color{blue}{\sin kx}} \cdot \sin ky \]

   7. Applied rewrites 45.1%

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]
   8. Step-by-step derivation

   9. Applied rewrites 68.3%

      \[\leadsto \frac{\sin th}{\left|\sin kx\right|} \cdot \sin ky \]

   if 0.70799999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 80.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.99:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\right)\right) \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.708:\\ \;\;\;\;\frac{\sin th}{\left|\sin kx\right|} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-5)
   (* (/ (sin th) (sin kx)) (* (fma (* ky ky) -0.16666666666666666 1.0) 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)))) <= 2e-5) {
		tmp = (sin(th) / sin(kx)) * (fma((ky * ky), -0.16666666666666666, 1.0) * ky);
	} 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)))) <= 2e-5)
		tmp = Float64(Float64(sin(th) / sin(kx)) * Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky));
	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], 2e-5], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000016e-5

   1. Initial program 96.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 96.4

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]
   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 38.8

         \[\leadsto \frac{\sin th}{\color{blue}{\sin kx}} \cdot \sin ky \]

   7. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\sin kx} \cdot \color{blue}{\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sin kx} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\sin kx} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th}{\sin kx} \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sin kx} \cdot \left(\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\sin kx} \cdot \left(\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin th}{\sin kx} \cdot \left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky\right) \]

      7. lower-*.f64 36.5

         \[\leadsto \frac{\sin th}{\sin kx} \cdot \left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky\right) \]

   10. Applied rewrites 36.5%

       \[\leadsto \frac{\sin th}{\sin kx} \cdot \color{blue}{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)} \]

   if 2.00000000000000016e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 67.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 45.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-5)
   (* (/ 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)))) <= 2e-5) {
		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)))) <= 2d-5) 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)))) <= 2e-5) {
		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)))) <= 2e-5:
		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)))) <= 2e-5)
		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)))) <= 2e-5)
		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], 2e-5], 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))))) < 2.00000000000000016e-5

   1. Initial program 96.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in 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 36.8

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 2.00000000000000016e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 67.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-5)
   (/ (* (sin th) ky) (sin kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-5) {
		tmp = (sin(th) * ky) / sin(kx);
	} 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)))) <= 2d-5) then
        tmp = (sin(th) * ky) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 2e-5) {
		tmp = (Math.sin(th) * ky) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 2e-5:
		tmp = (math.sin(th) * ky) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-5)
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-5)
		tmp = (sin(th) * ky) / sin(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-5], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $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))))) < 2.00000000000000016e-5

   1. Initial program 96.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sin kx} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sin kx} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th} \cdot ky}{\sin kx} \]

      5. lower-sin.f64 35.2

         \[\leadsto \frac{\sin th \cdot ky}{\color{blue}{\sin kx}} \]

   7. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   if 2.00000000000000016e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 67.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 16.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 10^{-296}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;1 \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      1e-296)
   (* (* (* -0.16666666666666666 th) th) th)
   (* 1.0 th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 1e-296) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = 1.0 * 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)))) * sin(th)) <= 1d-296) then
        tmp = (((-0.16666666666666666d0) * th) * th) * th
    else
        tmp = 1.0d0 * th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th)) <= 1e-296) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = 1.0 * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)) <= 1e-296:
		tmp = ((-0.16666666666666666 * th) * th) * th
	else:
		tmp = 1.0 * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-296)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = Float64(1.0 * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-296)
		tmp = ((-0.16666666666666666 * th) * th) * th;
	else
		tmp = 1.0 * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 1e-296], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[(1.0 * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 1e-296

   1. Initial program 96.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 27.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\sin th} \]

   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 18.3%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   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 16.2%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 16.2%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 1e-296 < (*.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 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 26.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\sin th} \]

   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 14.5%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 3.5%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   12. Taylor expanded in th around 0

       \[\leadsto 1 \cdot th \]
   13. Step-by-step derivation

   14. Applied rewrites 14.8%

       \[\leadsto 1 \cdot th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 31.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-95}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-95)
   (* (* (* -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)))) <= 4e-95) {
		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)))) <= 4d-95) 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)))) <= 4e-95) {
		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)))) <= 4e-95:
		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)))) <= 4e-95)
		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)))) <= 4e-95)
		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], 4e-95], 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))))) < 3.99999999999999996e-95

   1. Initial program 96.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\sin th} \]

   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 3.3%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 15.4%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 15.4%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 3.99999999999999996e-95 < (/.f64 (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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 60.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 72.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;\sin ky \leq -0.26:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\right)\right) \cdot \sin ky\\ \mathbf{elif}\;\sin ky \leq 0.0002:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= (sin ky) -0.26)
     (*
      (*
       (sqrt (pow (- 1.0 (cos (* -2.0 ky))) -1.0))
       (*
        (sqrt 2.0)
        (*
         (fma
          (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
          (* th th)
          1.0)
         th)))
      (sin ky))
     (if (<= (sin ky) 0.0002)
       (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (sin(ky) <= -0.26) {
		tmp = (sqrt(pow((1.0 - cos((-2.0 * ky))), -1.0)) * (sqrt(2.0) * (fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th))) * sin(ky);
	} else if (sin(ky) <= 0.0002) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (sin(ky) <= -0.26)
		tmp = Float64(Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * ky))) ^ -1.0)) * Float64(sqrt(2.0) * Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th))) * sin(ky));
	elseif (sin(ky) <= 0.0002)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.26], N[(N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.0002], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.26000000000000001

   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 99.7

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin ky \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin ky \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin ky \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin ky \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin ky \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   7. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\left(\left(\sin th \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}\right)} \cdot \sin ky \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \cdot \sin ky \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \cdot \sin ky \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      6. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      10. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(\color{blue}{-2} \cdot ky\right)}} \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sin ky \]

      11. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \color{blue}{\left(\sqrt{2} \cdot \sin th\right)}\right) \cdot \sin ky \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \color{blue}{\left(\sqrt{2} \cdot \sin th\right)}\right) \cdot \sin ky \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\color{blue}{\sqrt{2}} \cdot \sin th\right)\right) \cdot \sin ky \]

      14. lower-sin.f64 52.5

          \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \color{blue}{\sin th}\right)\right) \cdot \sin ky \]

   9. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \sin th\right)\right)} \cdot \sin ky \]

   10. Taylor expanded in th around 0

       \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \left(th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \cdot \sin ky \]
   11. Step-by-step derivation

   12. Applied rewrites 23.1%

       \[\leadsto \left(\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot \color{blue}{th}\right)\right)\right) \cdot \sin ky \]

   if -0.26000000000000001 < (sin.f64 ky) < 2.0000000000000001e-4

   1. Initial program 91.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-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 90.4

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 90.4%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 93.0

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   10. Applied rewrites 93.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   if 2.0000000000000001e-4 < (sin.f64 ky)

   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 65.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.26:\\ \;\;\;\;\left(\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\right)\right) \cdot \sin ky\\ \mathbf{elif}\;\sin ky \leq 0.0002:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 77.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;ky \leq 51:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= ky 51.0)
     (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
     (*
      (/
       (sin th)
       (/
        (sqrt
         (fma (- 1.0 (cos (* 2.0 ky))) 2.0 (* 2.0 (- 1.0 (cos (* 2.0 kx))))))
        2.0))
      (sin ky)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (ky <= 51.0) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else {
		tmp = (sin(th) / (sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, (2.0 * (1.0 - cos((2.0 * kx)))))) / 2.0)) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (ky <= 51.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) / 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(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[ky, 51.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $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[ky], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 51

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 65.1

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 65.1%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 68.0

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   10. Applied rewrites 68.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   if 51 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.5

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      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 99.5

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin ky \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin ky \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin ky \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin ky \]

      8. frac-add N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin ky \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin ky \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin ky \]

      11. sqrt-div N/A

          \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin ky \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{\color{blue}{4}}}} \cdot \sin ky \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\color{blue}{2}}} \cdot \sin ky \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{2}}} \cdot \sin ky \]

   6. Applied rewrites 99.0%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin ky \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 78.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;ky \leq 51:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= ky 51.0)
     (* (/ t_1 (hypot t_1 (sin 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 t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (ky <= 51.0) {
		tmp = (t_1 / hypot(t_1, sin(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)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (ky <= 51.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(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_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[ky, 51.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[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 ky < 51

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 65.1

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 65.1%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 68.0

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   10. Applied rewrites 68.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   if 51 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. 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 98.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;ky \leq 0.095:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)}} \cdot \left(\sqrt{2} \cdot \sin ky\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= ky 0.095)
     (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
     (*
      (/
       (sin th)
       (sqrt (+ (- 1.0 (cos (* -2.0 kx))) (- 1.0 (cos (* -2.0 ky))))))
      (* (sqrt 2.0) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (ky <= 0.095) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else {
		tmp = (sin(th) / sqrt(((1.0 - cos((-2.0 * kx))) + (1.0 - cos((-2.0 * ky)))))) * (sqrt(2.0) * sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (ky <= 0.095)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) / sqrt(Float64(Float64(1.0 - cos(Float64(-2.0 * kx))) + Float64(1.0 - cos(Float64(-2.0 * ky)))))) * Float64(sqrt(2.0) * sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[ky, 0.095], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.095000000000000001

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. 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 65.6

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 65.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 67.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   10. Applied rewrites 67.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   if 0.095000000000000001 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lower-*.f64 99.6

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      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 99.6

          \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   6. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)}} \cdot \left(\sqrt{2} \cdot \sin ky\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;ky \leq 0.095:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sin th}{\sqrt{\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)}} \cdot \sqrt{2}\right) \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= ky 0.095)
     (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
     (*
      (*
       (/
        (sin th)
        (sqrt (+ (- 1.0 (cos (* -2.0 kx))) (- 1.0 (cos (* -2.0 ky))))))
       (sqrt 2.0))
      (sin ky)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (ky <= 0.095) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else {
		tmp = ((sin(th) / sqrt(((1.0 - cos((-2.0 * kx))) + (1.0 - cos((-2.0 * ky)))))) * sqrt(2.0)) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (ky <= 0.095)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(Float64(sin(th) / sqrt(Float64(Float64(1.0 - cos(Float64(-2.0 * kx))) + Float64(1.0 - cos(Float64(-2.0 * ky)))))) * sqrt(2.0)) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[ky, 0.095], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.095000000000000001

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. 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 65.6

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 65.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 67.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   10. Applied rewrites 67.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   if 0.095000000000000001 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.5

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      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 99.5

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin ky \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin ky \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin ky \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin ky \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin ky \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   6. Applied rewrites 98.7%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\left(\frac{\sin th}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}\right)} \cdot \sin ky \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\sin th}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}\right)} \cdot \sin ky \]

   8. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(\frac{\sin th}{\sqrt{\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)}} \cdot \sqrt{2}\right)} \cdot \sin ky \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;ky \leq 0.095:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sin ky \cdot \frac{\sin th}{\sqrt{\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)}}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= ky 0.095)
     (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
     (*
      (*
       (sin ky)
       (/
        (sin th)
        (sqrt (+ (- 1.0 (cos (* -2.0 kx))) (- 1.0 (cos (* -2.0 ky)))))))
      (sqrt 2.0)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (ky <= 0.095) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else {
		tmp = (sin(ky) * (sin(th) / sqrt(((1.0 - cos((-2.0 * kx))) + (1.0 - cos((-2.0 * ky))))))) * sqrt(2.0);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (ky <= 0.095)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) * Float64(sin(th) / sqrt(Float64(Float64(1.0 - cos(Float64(-2.0 * kx))) + Float64(1.0 - cos(Float64(-2.0 * ky))))))) * sqrt(2.0));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[ky, 0.095], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.095000000000000001

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. 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 65.6

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 65.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 67.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   10. Applied rewrites 67.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   if 0.095000000000000001 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lower-*.f64 99.6

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      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 99.6

          \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   6. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{\sin th}{\sqrt{\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)}}\right) \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 68.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;ky \leq 1.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\left(\sqrt{2} \cdot \sin th\right) \cdot \sin ky\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= ky 1.25e+15)
     (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
     (*
      (sqrt (pow (- 1.0 (cos (* -2.0 ky))) -1.0))
      (* (* (sqrt 2.0) (sin th)) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (ky <= 1.25e+15) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else {
		tmp = sqrt(pow((1.0 - cos((-2.0 * ky))), -1.0)) * ((sqrt(2.0) * sin(th)) * sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (ky <= 1.25e+15)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	else
		tmp = Float64(sqrt((Float64(1.0 - cos(Float64(-2.0 * ky))) ^ -1.0)) * Float64(Float64(sqrt(2.0) * sin(th)) * sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[ky, 1.25e+15], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.25e15

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-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. 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 64.3

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 64.3%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 68.1

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   10. Applied rewrites 68.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   if 1.25e15 < ky

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.5

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      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 99.5

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin ky \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin ky \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin ky \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin ky \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin ky \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin ky \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   6. Applied rewrites 98.7%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin ky \]

   7. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot ky\right)}}} \cdot \left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      6. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}}} \cdot \left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}}} \cdot \left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}}} \cdot \left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}}} \cdot \left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(\color{blue}{-2} \cdot ky\right)}} \cdot \left(\sin ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \color{blue}{\left(\left(\sin th \cdot \sqrt{2}\right) \cdot \sin ky\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \color{blue}{\left(\left(\sin th \cdot \sqrt{2}\right) \cdot \sin ky\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sin th\right)} \cdot \sin ky\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sin th\right)} \cdot \sin ky\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\left(\color{blue}{\sqrt{2}} \cdot \sin th\right) \cdot \sin ky\right) \]

      16. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\left(\sqrt{2} \cdot \color{blue}{\sin th}\right) \cdot \sin ky\right) \]

      17. lower-sin.f64 55.4

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\left(\sqrt{2} \cdot \sin th\right) \cdot \color{blue}{\sin ky}\right) \]

   9. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot ky\right)}} \cdot \left(\left(\sqrt{2} \cdot \sin th\right) \cdot \sin ky\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 1.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(1 - \cos \left(-2 \cdot ky\right)\right)}^{-1}} \cdot \left(\left(\sqrt{2} \cdot \sin th\right) \cdot \sin ky\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 13.6% accurate, 105.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* 1.0 th))

C

double code(double kx, double ky, double th) {
	return 1.0 * th;
}

Fortran

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 = 1.0d0 * th
end function

Java

public static double code(double kx, double ky, double th) {
	return 1.0 * th;
}

Python

def code(kx, ky, th):
	return 1.0 * th

Julia

function code(kx, ky, th)
	return Float64(1.0 * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 1.0 * th;
end

Wolfram

code[kx_, ky_, th_] := N[(1.0 * th), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 27.1

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 27.1%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 16.6%

   \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

9. Taylor expanded in th around inf

   \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
10. Step-by-step derivation

11. Applied rewrites 10.6%

    \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

12. Taylor expanded in th around 0

    \[\leadsto 1 \cdot th \]
13. Step-by-step derivation

14. Applied rewrites 16.9%

    \[\leadsto 1 \cdot th \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(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)))