Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 6.3s

Alternatives: 23

Speedup: 1.2×

Specification

Math

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

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: 93.7% accurate, 1.0× speedup

Math

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

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

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

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 93.7%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7%

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

3. Applied rewrites 99.7%

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

Alternative 2: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := {t\_2}^{2}\\ t_4 := \sqrt{{\sin kx}^{2} + t\_3}\\ t_5 := \frac{t\_2}{t\_4}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_2\right)\\ \mathbf{elif}\;t\_5 \leq -0.125:\\ \;\;\;\;\frac{th \cdot t\_2}{t\_4}\\ \mathbf{elif}\;t\_5 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{t\_2}}\\ \mathbf{elif}\;t\_5 \leq 0.998:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot t\_2}{\mathsf{hypot}\left(t\_2, \sin kx\right)}\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\frac{t\_2}{\sqrt{{kx}^{2} + t\_3}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1
        (*
         (fabs ky)
         (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0)))))
       (t_2 (sin (fabs ky)))
       (t_3 (pow t_2 2.0))
       (t_4 (sqrt (+ (pow (sin kx) 2.0) t_3)))
       (t_5 (/ t_2 t_4)))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -1.0)
     (* (sin th) (copysign 1.0 t_2))
     (if (<= t_5 -0.125)
       (/ (* th t_2) t_4)
       (if (<= t_5 0.05)
         (/ (sin th) (/ (fabs (sin kx)) t_2))
         (if (<= t_5 0.998)
           (/
            (* (* (fma (* th th) -0.16666666666666666 1.0) th) t_2)
            (hypot t_2 (sin kx)))
           (if (<= t_5 2.0)
             (* (/ t_2 (sqrt (+ (pow kx 2.0) t_3))) (sin th))
             (* (/ t_1 (hypot t_1 (sin kx))) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double t_2 = sin(fabs(ky));
	double t_3 = pow(t_2, 2.0);
	double t_4 = sqrt((pow(sin(kx), 2.0) + t_3));
	double t_5 = t_2 / t_4;
	double tmp;
	if (t_5 <= -1.0) {
		tmp = sin(th) * copysign(1.0, t_2);
	} else if (t_5 <= -0.125) {
		tmp = (th * t_2) / t_4;
	} else if (t_5 <= 0.05) {
		tmp = sin(th) / (fabs(sin(kx)) / t_2);
	} else if (t_5 <= 0.998) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * th) * t_2) / hypot(t_2, sin(kx));
	} else if (t_5 <= 2.0) {
		tmp = (t_2 / sqrt((pow(kx, 2.0) + t_3))) * sin(th);
	} else {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	t_2 = sin(abs(ky))
	t_3 = t_2 ^ 2.0
	t_4 = sqrt(Float64((sin(kx) ^ 2.0) + t_3))
	t_5 = Float64(t_2 / t_4)
	tmp = 0.0
	if (t_5 <= -1.0)
		tmp = Float64(sin(th) * copysign(1.0, t_2));
	elseif (t_5 <= -0.125)
		tmp = Float64(Float64(th * t_2) / t_4);
	elseif (t_5 <= 0.05)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / t_2));
	elseif (t_5 <= 0.998)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * t_2) / hypot(t_2, sin(kx)));
	elseif (t_5 <= 2.0)
		tmp = Float64(Float64(t_2 / sqrt(Float64((kx ^ 2.0) + t_3))) * sin(th));
	else
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, -1.0], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -0.125], N[(N[(th * t$95$2), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 0.05], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.998], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * t$95$2), $MachinePrecision] / N[Sqrt[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[(t$95$2 / N[Sqrt[N[(N[Power[kx, 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.4%

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

   4. Applied rewrites 41.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. sqrt-prod N/A

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

      10. lower-unsound-*.f64 N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-unsound-sqrt.f64 30.6%

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

   6. Applied rewrites 30.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-fabs-rev N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-unprod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 44.2%

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

   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))))) < -0.125

   1. Initial program 93.7%

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

   2. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 46.1%

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

   4. Applied rewrites 46.1%

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

   if -0.125 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. associate-*l/ N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.3%

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

   8. Applied rewrites 41.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

      5. lift-fabs.f64 44.6%

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

   10. Applied rewrites 44.6%

       \[\leadsto \frac{\sin th}{\frac{\color{blue}{\left|\sin kx\right|}}{\sin ky}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.6%

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

   6. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 46.9%

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

   if 0.998 < (/.f64 (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 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 52.4%

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

   4. Applied rewrites 52.4%

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

   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 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.3%

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.7%

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

   9. Applied rewrites 55.7%

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := {t\_2}^{2}\\ t_4 := \sqrt{{\sin kx}^{2} + t\_3}\\ t_5 := \frac{t\_2}{t\_4}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_2\right)\\ \mathbf{elif}\;t\_5 \leq -0.125:\\ \;\;\;\;\frac{th}{t\_4} \cdot t\_2\\ \mathbf{elif}\;t\_5 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{t\_2}}\\ \mathbf{elif}\;t\_5 \leq 0.998:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot t\_2}{\mathsf{hypot}\left(t\_2, \sin kx\right)}\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\frac{t\_2}{\sqrt{{kx}^{2} + t\_3}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1
        (*
         (fabs ky)
         (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0)))))
       (t_2 (sin (fabs ky)))
       (t_3 (pow t_2 2.0))
       (t_4 (sqrt (+ (pow (sin kx) 2.0) t_3)))
       (t_5 (/ t_2 t_4)))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -1.0)
     (* (sin th) (copysign 1.0 t_2))
     (if (<= t_5 -0.125)
       (* (/ th t_4) t_2)
       (if (<= t_5 0.05)
         (/ (sin th) (/ (fabs (sin kx)) t_2))
         (if (<= t_5 0.998)
           (/
            (* (* (fma (* th th) -0.16666666666666666 1.0) th) t_2)
            (hypot t_2 (sin kx)))
           (if (<= t_5 2.0)
             (* (/ t_2 (sqrt (+ (pow kx 2.0) t_3))) (sin th))
             (* (/ t_1 (hypot t_1 (sin kx))) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double t_2 = sin(fabs(ky));
	double t_3 = pow(t_2, 2.0);
	double t_4 = sqrt((pow(sin(kx), 2.0) + t_3));
	double t_5 = t_2 / t_4;
	double tmp;
	if (t_5 <= -1.0) {
		tmp = sin(th) * copysign(1.0, t_2);
	} else if (t_5 <= -0.125) {
		tmp = (th / t_4) * t_2;
	} else if (t_5 <= 0.05) {
		tmp = sin(th) / (fabs(sin(kx)) / t_2);
	} else if (t_5 <= 0.998) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * th) * t_2) / hypot(t_2, sin(kx));
	} else if (t_5 <= 2.0) {
		tmp = (t_2 / sqrt((pow(kx, 2.0) + t_3))) * sin(th);
	} else {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	t_2 = sin(abs(ky))
	t_3 = t_2 ^ 2.0
	t_4 = sqrt(Float64((sin(kx) ^ 2.0) + t_3))
	t_5 = Float64(t_2 / t_4)
	tmp = 0.0
	if (t_5 <= -1.0)
		tmp = Float64(sin(th) * copysign(1.0, t_2));
	elseif (t_5 <= -0.125)
		tmp = Float64(Float64(th / t_4) * t_2);
	elseif (t_5 <= 0.05)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / t_2));
	elseif (t_5 <= 0.998)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * t_2) / hypot(t_2, sin(kx)));
	elseif (t_5 <= 2.0)
		tmp = Float64(Float64(t_2 / sqrt(Float64((kx ^ 2.0) + t_3))) * sin(th));
	else
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$4), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, -1.0], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -0.125], N[(N[(th / t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 0.05], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.998], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * t$95$2), $MachinePrecision] / N[Sqrt[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[(t$95$2 / N[Sqrt[N[(N[Power[kx, 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.4%

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

   4. Applied rewrites 41.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. sqrt-prod N/A

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

      10. lower-unsound-*.f64 N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-unsound-sqrt.f64 30.6%

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

   6. Applied rewrites 30.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-fabs-rev N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-unprod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 44.2%

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

   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))))) < -0.125

   1. Initial program 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 48.0%

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

   6. Applied rewrites 48.0%

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

   if -0.125 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. associate-*l/ N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.3%

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

   8. Applied rewrites 41.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

      5. lift-fabs.f64 44.6%

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

   10. Applied rewrites 44.6%

       \[\leadsto \frac{\sin th}{\frac{\color{blue}{\left|\sin kx\right|}}{\sin ky}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.6%

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

   6. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 46.9%

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

   if 0.998 < (/.f64 (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 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 52.4%

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

   4. Applied rewrites 52.4%

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

   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 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.3%

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.7%

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

   9. Applied rewrites 55.7%

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \sqrt{{\sin kx}^{2} + {t\_2}^{2}}\\ t_4 := \frac{t\_2}{t\_3}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_2\right)\\ \mathbf{elif}\;t\_4 \leq -0.125:\\ \;\;\;\;\frac{th}{t\_3} \cdot t\_2\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{t\_2}}\\ \mathbf{elif}\;t\_4 \leq 0.9999999982247365:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot t\_2}{\mathsf{hypot}\left(t\_2, \sin kx\right)}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1
        (*
         (fabs ky)
         (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0)))))
       (t_2 (sin (fabs ky)))
       (t_3 (sqrt (+ (pow (sin kx) 2.0) (pow t_2 2.0))))
       (t_4 (/ t_2 t_3)))
  (*
   (copysign 1.0 ky)
   (if (<= t_4 -1.0)
     (* (sin th) (copysign 1.0 t_2))
     (if (<= t_4 -0.125)
       (* (/ th t_3) t_2)
       (if (<= t_4 0.05)
         (/ (sin th) (/ (fabs (sin kx)) t_2))
         (if (<= t_4 0.9999999982247365)
           (/
            (* (* (fma (* th th) -0.16666666666666666 1.0) th) t_2)
            (hypot t_2 (sin kx)))
           (if (<= t_4 2.0)
             (sin th)
             (* (/ t_1 (hypot t_1 (sin kx))) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double t_2 = sin(fabs(ky));
	double t_3 = sqrt((pow(sin(kx), 2.0) + pow(t_2, 2.0)));
	double t_4 = t_2 / t_3;
	double tmp;
	if (t_4 <= -1.0) {
		tmp = sin(th) * copysign(1.0, t_2);
	} else if (t_4 <= -0.125) {
		tmp = (th / t_3) * t_2;
	} else if (t_4 <= 0.05) {
		tmp = sin(th) / (fabs(sin(kx)) / t_2);
	} else if (t_4 <= 0.9999999982247365) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * th) * t_2) / hypot(t_2, sin(kx));
	} else if (t_4 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	t_2 = sin(abs(ky))
	t_3 = sqrt(Float64((sin(kx) ^ 2.0) + (t_2 ^ 2.0)))
	t_4 = Float64(t_2 / t_3)
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(sin(th) * copysign(1.0, t_2));
	elseif (t_4 <= -0.125)
		tmp = Float64(Float64(th / t_3) * t_2);
	elseif (t_4 <= 0.05)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / t_2));
	elseif (t_4 <= 0.9999999982247365)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * t_2) / hypot(t_2, sin(kx)));
	elseif (t_4 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / t$95$3), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, -1.0], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.125], N[(N[(th / t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.05], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9999999982247365], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * t$95$2), $MachinePrecision] / N[Sqrt[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[Sin[th], $MachinePrecision], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.4%

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

   4. Applied rewrites 41.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. sqrt-prod N/A

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

      10. lower-unsound-*.f64 N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-unsound-sqrt.f64 30.6%

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

   6. Applied rewrites 30.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-fabs-rev N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-unprod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 44.2%

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

   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))))) < -0.125

   1. Initial program 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 48.0%

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

   6. Applied rewrites 48.0%

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

   if -0.125 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. associate-*l/ N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.3%

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

   8. Applied rewrites 41.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

      5. lift-fabs.f64 44.6%

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

   10. Applied rewrites 44.6%

       \[\leadsto \frac{\sin th}{\frac{\color{blue}{\left|\sin kx\right|}}{\sin ky}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999999822473651

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.6%

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

   6. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 46.9%

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

   if 0.99999999822473651 < (/.f64 (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 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.4%

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

   4. Applied rewrites 41.4%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]

   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 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.3%

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.7%

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

   9. Applied rewrites 55.7%

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \sqrt{{\sin kx}^{2} + {t\_1}^{2}}\\ t_3 := \frac{t\_1}{t\_2}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_3 \leq -0.125:\\ \;\;\;\;\frac{th \cdot t\_1}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{t\_1}}\\ \mathbf{elif}\;t\_3 \leq 0.998:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(kx \cdot \left(1 + -0.16666666666666666 \cdot {kx}^{2}\right), t\_1\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))
       (t_3 (/ t_1 t_2)))
  (*
   (copysign 1.0 ky)
   (if (<= t_3 -1.0)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_3 -0.125)
       (/ (* th t_1) t_2)
       (if (<= t_3 0.05)
         (/ (sin th) (/ (fabs (sin kx)) t_1))
         (if (<= t_3 0.998)
           (/
            (* (* (fma (* th th) -0.16666666666666666 1.0) th) t_1)
            (hypot t_1 (sin kx)))
           (*
            (/
             (sin th)
             (hypot
              (* kx (+ 1.0 (* -0.16666666666666666 (pow kx 2.0))))
              t_1))
            t_1))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -1.0) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_3 <= -0.125) {
		tmp = (th * t_1) / t_2;
	} else if (t_3 <= 0.05) {
		tmp = sin(th) / (fabs(sin(kx)) / t_1);
	} else if (t_3 <= 0.998) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * th) * t_1) / hypot(t_1, sin(kx));
	} else {
		tmp = (sin(th) / hypot((kx * (1.0 + (-0.16666666666666666 * pow(kx, 2.0)))), t_1)) * t_1;
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))
	t_3 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_3 <= -0.125)
		tmp = Float64(Float64(th * t_1) / t_2);
	elseif (t_3 <= 0.05)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / t_1));
	elseif (t_3 <= 0.998)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * t_1) / hypot(t_1, sin(kx)));
	else
		tmp = Float64(Float64(sin(th) / hypot(Float64(kx * Float64(1.0 + Float64(-0.16666666666666666 * (kx ^ 2.0)))), t_1)) * t_1);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -1.0], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.125], N[(N[(th * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.998], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(kx * N[(1.0 + N[(-0.16666666666666666 * N[Power[kx, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $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 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.4%

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

   4. Applied rewrites 41.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. sqrt-prod N/A

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

      10. lower-unsound-*.f64 N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-unsound-sqrt.f64 30.6%

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

   6. Applied rewrites 30.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-fabs-rev N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-unprod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 44.2%

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

   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))))) < -0.125

   1. Initial program 93.7%

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

   2. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 46.1%

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

   4. Applied rewrites 46.1%

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

   if -0.125 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. associate-*l/ N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.3%

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

   8. Applied rewrites 41.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

      5. lift-fabs.f64 44.6%

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

   10. Applied rewrites 44.6%

       \[\leadsto \frac{\sin th}{\frac{\color{blue}{\left|\sin kx\right|}}{\sin ky}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.6%

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

   6. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 46.9%

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

   if 0.998 < (/.f64 (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 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 58.2%

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

   6. Applied rewrites 58.2%

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx \cdot \left(1 + -0.16666666666666666 \cdot {kx}^{2}\right)}, \sin ky\right)} \cdot \sin ky \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \frac{t\_2}{\sqrt{{\sin kx}^{2} + {t\_2}^{2}}}\\ t_4 := \mathsf{hypot}\left(t\_2, \sin kx\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_2\right)\\ \mathbf{elif}\;t\_3 \leq -0.125:\\ \;\;\;\;\frac{t\_2}{t\_4} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, -0.16666666666666666, th\right)\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{t\_2}}\\ \mathbf{elif}\;t\_3 \leq 0.9999999982247365:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot t\_2}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1
        (*
         (fabs ky)
         (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0)))))
       (t_2 (sin (fabs ky)))
       (t_3 (/ t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_2 2.0)))))
       (t_4 (hypot t_2 (sin kx))))
  (*
   (copysign 1.0 ky)
   (if (<= t_3 -1.0)
     (* (sin th) (copysign 1.0 t_2))
     (if (<= t_3 -0.125)
       (* (/ t_2 t_4) (fma (* (* th th) th) -0.16666666666666666 th))
       (if (<= t_3 0.05)
         (/ (sin th) (/ (fabs (sin kx)) t_2))
         (if (<= t_3 0.9999999982247365)
           (/
            (* (* (fma (* th th) -0.16666666666666666 1.0) th) t_2)
            t_4)
           (if (<= t_3 2.0)
             (sin th)
             (* (/ t_1 (hypot t_1 (sin kx))) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double t_2 = sin(fabs(ky));
	double t_3 = t_2 / sqrt((pow(sin(kx), 2.0) + pow(t_2, 2.0)));
	double t_4 = hypot(t_2, sin(kx));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = sin(th) * copysign(1.0, t_2);
	} else if (t_3 <= -0.125) {
		tmp = (t_2 / t_4) * fma(((th * th) * th), -0.16666666666666666, th);
	} else if (t_3 <= 0.05) {
		tmp = sin(th) / (fabs(sin(kx)) / t_2);
	} else if (t_3 <= 0.9999999982247365) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * th) * t_2) / t_4;
	} else if (t_3 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	t_2 = sin(abs(ky))
	t_3 = Float64(t_2 / sqrt(Float64((sin(kx) ^ 2.0) + (t_2 ^ 2.0))))
	t_4 = hypot(t_2, sin(kx))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(sin(th) * copysign(1.0, t_2));
	elseif (t_3 <= -0.125)
		tmp = Float64(Float64(t_2 / t_4) * fma(Float64(Float64(th * th) * th), -0.16666666666666666, th));
	elseif (t_3 <= 0.05)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / t_2));
	elseif (t_3 <= 0.9999999982247365)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * t_2) / t_4);
	elseif (t_3 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -1.0], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$2]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.125], N[(N[(t$95$2 / t$95$4), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666 + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9999999982247365], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[Sin[th], $MachinePrecision], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.4%

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

   4. Applied rewrites 41.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. sqrt-prod N/A

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

      10. lower-unsound-*.f64 N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-unsound-sqrt.f64 30.6%

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

   6. Applied rewrites 30.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-fabs-rev N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-unprod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 44.2%

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

   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))))) < -0.125

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.6%

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

   6. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. cube-unmult N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. lift-pow.f64 N/A

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

      16. lower-*.f64 50.6%

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left({th}^{2} \cdot th, -0.16666666666666666, th\right) \]

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 50.6%

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

   8. Applied rewrites 50.6%

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

   if -0.125 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. associate-*l/ N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.3%

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

   8. Applied rewrites 41.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

      5. lift-fabs.f64 44.6%

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

   10. Applied rewrites 44.6%

       \[\leadsto \frac{\sin th}{\frac{\color{blue}{\left|\sin kx\right|}}{\sin ky}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999999822473651

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.6%

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

   6. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 46.9%

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

   if 0.99999999822473651 < (/.f64 (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 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.4%

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

   4. Applied rewrites 41.4%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]

   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 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.3%

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.7%

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

   9. Applied rewrites 55.7%

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 7: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := \mathsf{hypot}\left(t\_1, \sin kx\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -0.125:\\ \;\;\;\;\frac{t\_1}{t\_3} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, -0.16666666666666666, th\right)\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{t\_1}}\\ \mathbf{elif}\;t\_2 \leq 0.9999999982247365:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot t\_1}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{t\_3 \cdot \frac{1}{\left|ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
       (t_3 (hypot t_1 (sin kx))))
  (*
   (copysign 1.0 ky)
   (if (<= t_2 -1.0)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_2 -0.125)
       (* (/ t_1 t_3) (fma (* (* th th) th) -0.16666666666666666 th))
       (if (<= t_2 0.05)
         (/ (sin th) (/ (fabs (sin kx)) t_1))
         (if (<= t_2 0.9999999982247365)
           (/
            (* (* (fma (* th th) -0.16666666666666666 1.0) th) t_1)
            t_3)
           (if (<= t_2 2.0)
             (sin th)
             (/ (sin th) (* t_3 (/ 1.0 (fabs ky))))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = hypot(t_1, sin(kx));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_2 <= -0.125) {
		tmp = (t_1 / t_3) * fma(((th * th) * th), -0.16666666666666666, th);
	} else if (t_2 <= 0.05) {
		tmp = sin(th) / (fabs(sin(kx)) / t_1);
	} else if (t_2 <= 0.9999999982247365) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * th) * t_1) / t_3;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (t_3 * (1.0 / fabs(ky)));
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_3 = hypot(t_1, sin(kx))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_2 <= -0.125)
		tmp = Float64(Float64(t_1 / t_3) * fma(Float64(Float64(th * th) * th), -0.16666666666666666, th));
	elseif (t_2 <= 0.05)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / t_1));
	elseif (t_2 <= 0.9999999982247365)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * t_1) / t_3);
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(t_3 * Float64(1.0 / abs(ky))));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1.0], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.125], N[(N[(t$95$1 / t$95$3), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666 + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.05], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999999982247365], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(t$95$3 * N[(1.0 / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.4%

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

   4. Applied rewrites 41.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. sqrt-prod N/A

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

      10. lower-unsound-*.f64 N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. cos-diff N/A

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

      13. cos-sin-sum N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-unsound-sqrt.f64 30.6%

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

   6. Applied rewrites 30.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-fabs-rev N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. sqrt-unprod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 44.2%

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

   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))))) < -0.125

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.6%

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

   6. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. cube-unmult N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. lift-pow.f64 N/A

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

      16. lower-*.f64 50.6%

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left({th}^{2} \cdot th, -0.16666666666666666, th\right) \]

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 50.6%

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

   8. Applied rewrites 50.6%

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

   if -0.125 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      3. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

   8. Applied rewrites 41.3%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx}}{\sin ky}} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

      5. lift-fabs.f64 44.6%

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

   10. Applied rewrites 44.6%

       \[\leadsto \frac{\sin th}{\frac{\color{blue}{\left|\sin kx\right|}}{\sin ky}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999999822473651

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right)\right) \]

      4. lower-pow.f64 50.6%

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{\color{blue}{2}}\right)\right) \]

   6. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   8. Applied rewrites 46.9%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   if 0.99999999822473651 < (/.f64 (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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]

   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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{1}{\sin ky}}} \]

      3. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \frac{1}{\sin ky}} \]

      4. pow1/2 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{{\left(\sin kx \cdot \sin kx + \sin ky \cdot \sin ky\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sin ky}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\sin th}{{\color{blue}{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)}}^{\frac{1}{2}} \cdot \frac{1}{\sin ky}} \]

      6. pow1/2 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \frac{1}{\sin ky}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \frac{1}{\sin ky}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}} \]

      9. lower-/.f64 99.5%

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{\sin ky}}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{ky}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.5%

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\color{blue}{ky}}} \]

   10. Applied rewrites 51.5%

       \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{ky}}} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 8: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := \mathsf{hypot}\left(t\_1, \sin kx\right)\\ t_4 := \frac{t\_1}{t\_3} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, -0.16666666666666666, th\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -0.125:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{t\_1}}\\ \mathbf{elif}\;t\_2 \leq 0.9999999982247365:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{t\_3 \cdot \frac{1}{\left|ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
       (t_3 (hypot t_1 (sin kx)))
       (t_4
        (*
         (/ t_1 t_3)
         (fma (* (* th th) th) -0.16666666666666666 th))))
  (*
   (copysign 1.0 ky)
   (if (<= t_2 -1.0)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_2 -0.125)
       t_4
       (if (<= t_2 0.05)
         (/ (sin th) (/ (fabs (sin kx)) t_1))
         (if (<= t_2 0.9999999982247365)
           t_4
           (if (<= t_2 2.0)
             (sin th)
             (/ (sin th) (* t_3 (/ 1.0 (fabs ky))))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = hypot(t_1, sin(kx));
	double t_4 = (t_1 / t_3) * fma(((th * th) * th), -0.16666666666666666, th);
	double tmp;
	if (t_2 <= -1.0) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_2 <= -0.125) {
		tmp = t_4;
	} else if (t_2 <= 0.05) {
		tmp = sin(th) / (fabs(sin(kx)) / t_1);
	} else if (t_2 <= 0.9999999982247365) {
		tmp = t_4;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (t_3 * (1.0 / fabs(ky)));
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_3 = hypot(t_1, sin(kx))
	t_4 = Float64(Float64(t_1 / t_3) * fma(Float64(Float64(th * th) * th), -0.16666666666666666, th))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_2 <= -0.125)
		tmp = t_4;
	elseif (t_2 <= 0.05)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / t_1));
	elseif (t_2 <= 0.9999999982247365)
		tmp = t_4;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(t_3 * Float64(1.0 / abs(ky))));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 / t$95$3), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666 + th), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1.0], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.125], t$95$4, If[LessEqual[t$95$2, 0.05], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999999982247365], t$95$4, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(t$95$3 * N[(1.0 / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      6. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}}} \]

      7. mult-flip N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\color{blue}{\frac{1}{2}}}} \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      17. lower-unsound-sqrt.f64 30.6%

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{0.5}} \]

   6. Applied rewrites 30.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{0.5}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{1 - \cos \left(ky + ky\right)}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{1 - \cos \left(ky + ky\right)}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \cos \left(ky + ky\right)}} \]

      5. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}}} \]

      8. sqrt-unprod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

   8. Applied rewrites 44.2%

      \[\leadsto \sin th \cdot \color{blue}{\mathsf{copysign}\left(1, \sin ky\right)} \]

   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))))) < -0.125 or 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999999822473651

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right)\right) \]

      4. lower-pow.f64 50.6%

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{\color{blue}{2}}\right)\right) \]

   6. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}\right) \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2} + \color{blue}{1}\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th \cdot 1}\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left({th}^{2} \cdot \frac{-1}{6}\right) + th\right) \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left(th \cdot {th}^{2}\right) \cdot \frac{-1}{6} + th\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th \cdot {th}^{2}, \color{blue}{\frac{-1}{6}}, th\right) \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th \cdot {th}^{2}, \frac{-1}{6}, th\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th \cdot \left(th \cdot th\right), \frac{-1}{6}, th\right) \]

      12. cube-unmult N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left({th}^{3}, \frac{-1}{6}, th\right) \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left({th}^{\left(2 + 1\right)}, \frac{-1}{6}, th\right) \]

      14. pow-plus N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left({th}^{2} \cdot th, \frac{-1}{6}, th\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left({th}^{2} \cdot th, \frac{-1}{6}, th\right) \]

      16. lower-*.f64 50.6%

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left({th}^{2} \cdot th, -0.16666666666666666, th\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left({th}^{2} \cdot th, \frac{-1}{6}, th\right) \]

      18. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, \frac{-1}{6}, th\right) \]

      19. lower-*.f64 50.6%

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, -0.16666666666666666, th\right) \]

   8. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, \color{blue}{-0.16666666666666666}, th\right) \]

   if -0.125 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      3. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

   8. Applied rewrites 41.3%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx}}{\sin ky}} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

      5. lift-fabs.f64 44.6%

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

   10. Applied rewrites 44.6%

       \[\leadsto \frac{\sin th}{\frac{\color{blue}{\left|\sin kx\right|}}{\sin ky}} \]

   if 0.99999999822473651 < (/.f64 (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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]

   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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{1}{\sin ky}}} \]

      3. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \frac{1}{\sin ky}} \]

      4. pow1/2 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{{\left(\sin kx \cdot \sin kx + \sin ky \cdot \sin ky\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sin ky}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\sin th}{{\color{blue}{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)}}^{\frac{1}{2}} \cdot \frac{1}{\sin ky}} \]

      6. pow1/2 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \frac{1}{\sin ky}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \frac{1}{\sin ky}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}} \]

      9. lower-/.f64 99.5%

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{\sin ky}}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{ky}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.5%

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\color{blue}{ky}}} \]

   10. Applied rewrites 51.5%

       \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{ky}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \mathsf{hypot}\left(t\_1, \sin kx\right)\\ t_3 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_4 := t\_1 \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{t\_2}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_3 \leq -0.125:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{t\_1}}\\ \mathbf{elif}\;t\_3 \leq 0.9999999982247365:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{t\_2 \cdot \frac{1}{\left|ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (hypot t_1 (sin kx)))
       (t_3 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
       (t_4
        (*
         t_1
         (/ (* (fma (* th th) -0.16666666666666666 1.0) th) t_2))))
  (*
   (copysign 1.0 ky)
   (if (<= t_3 -1.0)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_3 -0.125)
       t_4
       (if (<= t_3 0.05)
         (/ (sin th) (/ (fabs (sin kx)) t_1))
         (if (<= t_3 0.9999999982247365)
           t_4
           (if (<= t_3 2.0)
             (sin th)
             (/ (sin th) (* t_2 (/ 1.0 (fabs ky))))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = hypot(t_1, sin(kx));
	double t_3 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_4 = t_1 * ((fma((th * th), -0.16666666666666666, 1.0) * th) / t_2);
	double tmp;
	if (t_3 <= -1.0) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_3 <= -0.125) {
		tmp = t_4;
	} else if (t_3 <= 0.05) {
		tmp = sin(th) / (fabs(sin(kx)) / t_1);
	} else if (t_3 <= 0.9999999982247365) {
		tmp = t_4;
	} else if (t_3 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(th) / (t_2 * (1.0 / fabs(ky)));
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = hypot(t_1, sin(kx))
	t_3 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_4 = Float64(t_1 * Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / t_2))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_3 <= -0.125)
		tmp = t_4;
	elseif (t_3 <= 0.05)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / t_1));
	elseif (t_3 <= 0.9999999982247365)
		tmp = t_4;
	elseif (t_3 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) / Float64(t_2 * Float64(1.0 / abs(ky))));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -1.0], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.125], t$95$4, If[LessEqual[t$95$3, 0.05], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9999999982247365], t$95$4, If[LessEqual[t$95$3, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(t$95$2 * N[(1.0 / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      6. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}}} \]

      7. mult-flip N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\color{blue}{\frac{1}{2}}}} \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      17. lower-unsound-sqrt.f64 30.6%

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{0.5}} \]

   6. Applied rewrites 30.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{0.5}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{1 - \cos \left(ky + ky\right)}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{1 - \cos \left(ky + ky\right)}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \cos \left(ky + ky\right)}} \]

      5. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}}} \]

      8. sqrt-unprod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

   8. Applied rewrites 44.2%

      \[\leadsto \sin th \cdot \color{blue}{\mathsf{copysign}\left(1, \sin ky\right)} \]

   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))))) < -0.125 or 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999999822473651

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right)\right) \]

      4. lower-pow.f64 50.6%

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{\color{blue}{2}}\right)\right) \]

   6. Applied rewrites 50.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      6. lower-/.f64 50.6%

         \[\leadsto \sin ky \cdot \color{blue}{\frac{th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   8. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   if -0.125 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      3. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

   8. Applied rewrites 41.3%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx}}{\sin ky}} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

      5. lift-fabs.f64 44.6%

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

   10. Applied rewrites 44.6%

       \[\leadsto \frac{\sin th}{\frac{\color{blue}{\left|\sin kx\right|}}{\sin ky}} \]

   if 0.99999999822473651 < (/.f64 (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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]

   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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{1}{\sin ky}}} \]

      3. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \frac{1}{\sin ky}} \]

      4. pow1/2 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{{\left(\sin kx \cdot \sin kx + \sin ky \cdot \sin ky\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sin ky}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\sin th}{{\color{blue}{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)}}^{\frac{1}{2}} \cdot \frac{1}{\sin ky}} \]

      6. pow1/2 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \frac{1}{\sin ky}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \frac{1}{\sin ky}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}} \]

      9. lower-/.f64 99.5%

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{\sin ky}}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{ky}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.5%

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\color{blue}{ky}}} \]

   10. Applied rewrites 51.5%

       \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{ky}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\left|ky\right| \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(t\_1, \sin kx\right) \cdot \frac{1}{\left|ky\right|}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky))))
  (*
   (copysign 1.0 ky)
   (if (<= (fabs ky) 2.4e-5)
     (/ (sin th) (* (hypot t_1 (sin kx)) (/ 1.0 (fabs ky))))
     (* (sin th) (copysign 1.0 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if (fabs(ky) <= 2.4e-5) {
		tmp = sin(th) / (hypot(t_1, sin(kx)) * (1.0 / fabs(ky)));
	} else {
		tmp = sin(th) * copysign(1.0, t_1);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if (Math.abs(ky) <= 2.4e-5) {
		tmp = Math.sin(th) / (Math.hypot(t_1, Math.sin(kx)) * (1.0 / Math.abs(ky)));
	} else {
		tmp = Math.sin(th) * Math.copySign(1.0, t_1);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if math.fabs(ky) <= 2.4e-5:
		tmp = math.sin(th) / (math.hypot(t_1, math.sin(kx)) * (1.0 / math.fabs(ky)))
	else:
		tmp = math.sin(th) * math.copysign(1.0, t_1)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (abs(ky) <= 2.4e-5)
		tmp = Float64(sin(th) / Float64(hypot(t_1, sin(kx)) * Float64(1.0 / abs(ky))));
	else
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if (abs(ky) <= 2.4e-5)
		tmp = sin(th) / (hypot(t_1, sin(kx)) * (1.0 / abs(ky)));
	else
		tmp = sin(th) * (sign(t_1) * abs(1.0));
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[ky], $MachinePrecision], 2.4e-5], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] * N[(1.0 / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.4000000000000001e-5

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{1}{\sin ky}}} \]

      3. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \frac{1}{\sin ky}} \]

      4. pow1/2 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{{\left(\sin kx \cdot \sin kx + \sin ky \cdot \sin ky\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sin ky}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\sin th}{{\color{blue}{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)}}^{\frac{1}{2}} \cdot \frac{1}{\sin ky}} \]

      6. pow1/2 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \frac{1}{\sin ky}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \frac{1}{\sin ky}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}} \]

      9. lower-/.f64 99.5%

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{\sin ky}}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{ky}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.5%

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\color{blue}{ky}}} \]

   10. Applied rewrites 51.5%

       \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{ky}}} \]

   if 2.4000000000000001e-5 < ky

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      6. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}}} \]

      7. mult-flip N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\color{blue}{\frac{1}{2}}}} \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      17. lower-unsound-sqrt.f64 30.6%

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{0.5}} \]

   6. Applied rewrites 30.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{0.5}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{1 - \cos \left(ky + ky\right)}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{1 - \cos \left(ky + ky\right)}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \cos \left(ky + ky\right)}} \]

      5. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}}} \]

      8. sqrt-unprod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

   8. Applied rewrites 44.2%

      \[\leadsto \sin th \cdot \color{blue}{\mathsf{copysign}\left(1, \sin ky\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\left|ky\right| \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(t\_1, \sin kx\right) \cdot \frac{1}{\left|ky\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\sqrt{{t\_1}^{2}}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky))))
  (*
   (copysign 1.0 ky)
   (if (<= (fabs ky) 2.4e-5)
     (/ (sin th) (* (hypot t_1 (sin kx)) (/ 1.0 (fabs ky))))
     (* (/ t_1 (sqrt (pow t_1 2.0))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if (fabs(ky) <= 2.4e-5) {
		tmp = sin(th) / (hypot(t_1, sin(kx)) * (1.0 / fabs(ky)));
	} else {
		tmp = (t_1 / sqrt(pow(t_1, 2.0))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if (Math.abs(ky) <= 2.4e-5) {
		tmp = Math.sin(th) / (Math.hypot(t_1, Math.sin(kx)) * (1.0 / Math.abs(ky)));
	} else {
		tmp = (t_1 / Math.sqrt(Math.pow(t_1, 2.0))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if math.fabs(ky) <= 2.4e-5:
		tmp = math.sin(th) / (math.hypot(t_1, math.sin(kx)) * (1.0 / math.fabs(ky)))
	else:
		tmp = (t_1 / math.sqrt(math.pow(t_1, 2.0))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (abs(ky) <= 2.4e-5)
		tmp = Float64(sin(th) / Float64(hypot(t_1, sin(kx)) * Float64(1.0 / abs(ky))));
	else
		tmp = Float64(Float64(t_1 / sqrt((t_1 ^ 2.0))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if (abs(ky) <= 2.4e-5)
		tmp = sin(th) / (hypot(t_1, sin(kx)) * (1.0 / abs(ky)));
	else
		tmp = (t_1 / sqrt((t_1 ^ 2.0))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[ky], $MachinePrecision], 2.4e-5], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] * N[(1.0 / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.4000000000000001e-5

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

      2. mult-flip N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{1}{\sin ky}}} \]

      3. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \frac{1}{\sin ky}} \]

      4. pow1/2 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{{\left(\sin kx \cdot \sin kx + \sin ky \cdot \sin ky\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sin ky}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\sin th}{{\color{blue}{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)}}^{\frac{1}{2}} \cdot \frac{1}{\sin ky}} \]

      6. pow1/2 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \frac{1}{\sin ky}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \frac{1}{\sin ky}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}} \]

      9. lower-/.f64 99.5%

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{\sin ky}}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\sin ky}}} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{ky}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.5%

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{1}{\color{blue}{ky}}} \]

   10. Applied rewrites 51.5%

       \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\frac{1}{ky}}} \]

   if 2.4000000000000001e-5 < ky

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \cdot \sin th \]

      2. lower-sin.f64 40.6%

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 40.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 77.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.708:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

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 -0.708)
    (* (sin th) (copysign 1.0 (sin ky)))
    (if (<= t_1 0.7)
      (/ (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 <= -0.708) {
		tmp = sin(th) * copysign(1.0, sin(ky));
	} else if (t_1 <= 0.7) {
		tmp = sin(th) / (fabs(sin(kx)) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.708) {
		tmp = Math.sin(th) * Math.copySign(1.0, Math.sin(ky));
	} else if (t_1 <= 0.7) {
		tmp = Math.sin(th) / (Math.abs(Math.sin(kx)) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.708:
		tmp = math.sin(th) * math.copysign(1.0, math.sin(ky))
	elif t_1 <= 0.7:
		tmp = math.sin(th) / (math.fabs(math.sin(kx)) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.708)
		tmp = Float64(sin(th) * copysign(1.0, sin(ky)));
	elseif (t_1 <= 0.7)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.708)
		tmp = sin(th) * (sign(sin(ky)) * abs(1.0));
	elseif (t_1 <= 0.7)
		tmp = sin(th) / (abs(sin(kx)) / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.708], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[N[Sin[ky], $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.7], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.70799999999999996

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      6. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}}} \]

      7. mult-flip N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\color{blue}{\frac{1}{2}}}} \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      17. lower-unsound-sqrt.f64 30.6%

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{0.5}} \]

   6. Applied rewrites 30.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{0.5}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{1 - \cos \left(ky + ky\right)}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{1 - \cos \left(ky + ky\right)}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \cos \left(ky + ky\right)}} \]

      5. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}}} \]

      8. sqrt-unprod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

   8. Applied rewrites 44.2%

      \[\leadsto \sin th \cdot \color{blue}{\mathsf{copysign}\left(1, \sin ky\right)} \]

   if -0.70799999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      3. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

   8. Applied rewrites 41.3%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx}}{\sin ky}} \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

      5. lift-fabs.f64 44.6%

         \[\leadsto \frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}} \]

   10. Applied rewrites 44.6%

       \[\leadsto \frac{\sin th}{\frac{\color{blue}{\left|\sin kx\right|}}{\sin ky}} \]

   if 0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 77.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.708:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin th}{\left|\sin kx\right|} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

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 -0.708)
    (* (sin th) (copysign 1.0 (sin ky)))
    (if (<= t_1 0.7)
      (* (/ (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 <= -0.708) {
		tmp = sin(th) * copysign(1.0, sin(ky));
	} else if (t_1 <= 0.7) {
		tmp = (sin(th) / fabs(sin(kx))) * sin(ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.708) {
		tmp = Math.sin(th) * Math.copySign(1.0, Math.sin(ky));
	} else if (t_1 <= 0.7) {
		tmp = (Math.sin(th) / Math.abs(Math.sin(kx))) * Math.sin(ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.708:
		tmp = math.sin(th) * math.copysign(1.0, math.sin(ky))
	elif t_1 <= 0.7:
		tmp = (math.sin(th) / math.fabs(math.sin(kx))) * math.sin(ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.708)
		tmp = Float64(sin(th) * copysign(1.0, sin(ky)));
	elseif (t_1 <= 0.7)
		tmp = Float64(Float64(sin(th) / abs(sin(kx))) * sin(ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.708)
		tmp = sin(th) * (sign(sin(ky)) * abs(1.0));
	elseif (t_1 <= 0.7)
		tmp = (sin(th) / abs(sin(kx))) * sin(ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.708], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[N[Sin[ky], $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.7], 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))))) < -0.70799999999999996

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      6. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}}} \]

      7. mult-flip N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\color{blue}{\frac{1}{2}}}} \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      17. lower-unsound-sqrt.f64 30.6%

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{0.5}} \]

   6. Applied rewrites 30.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{0.5}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{1 - \cos \left(ky + ky\right)}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{1 - \cos \left(ky + ky\right)}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \cos \left(ky + ky\right)}} \]

      5. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}}} \]

      8. sqrt-unprod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

   8. Applied rewrites 44.2%

      \[\leadsto \sin th \cdot \color{blue}{\mathsf{copysign}\left(1, \sin ky\right)} \]

   if -0.70799999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      3. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

   8. Applied rewrites 41.3%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\color{blue}{\sin ky}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}}} \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2}}} \cdot \sin ky} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2}}} \cdot \sin ky} \]

   10. Applied rewrites 44.6%

       \[\leadsto \color{blue}{\frac{\sin th}{\left|\sin kx\right|} \cdot \sin ky} \]

   if 0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 77.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.708:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin ky}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

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 -0.708)
    (* (sin th) (copysign 1.0 (sin ky)))
    (if (<= t_1 0.7)
      (* (/ (sin ky) (fabs (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 <= -0.708) {
		tmp = sin(th) * copysign(1.0, sin(ky));
	} else if (t_1 <= 0.7) {
		tmp = (sin(ky) / fabs(sin(kx))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.708) {
		tmp = Math.sin(th) * Math.copySign(1.0, Math.sin(ky));
	} else if (t_1 <= 0.7) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(kx))) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.708:
		tmp = math.sin(th) * math.copysign(1.0, math.sin(ky))
	elif t_1 <= 0.7:
		tmp = (math.sin(ky) / math.fabs(math.sin(kx))) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.708)
		tmp = Float64(sin(th) * copysign(1.0, sin(ky)));
	elseif (t_1 <= 0.7)
		tmp = Float64(Float64(sin(ky) / abs(sin(kx))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.708)
		tmp = sin(th) * (sign(sin(ky)) * abs(1.0));
	elseif (t_1 <= 0.7)
		tmp = (sin(ky) / abs(sin(kx))) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.708], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[N[Sin[ky], $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.70799999999999996

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      6. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}}} \]

      7. mult-flip N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\color{blue}{\frac{1}{2}}}} \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      17. lower-unsound-sqrt.f64 30.6%

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{0.5}} \]

   6. Applied rewrites 30.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{0.5}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{1 - \cos \left(ky + ky\right)}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{1 - \cos \left(ky + ky\right)}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \cos \left(ky + ky\right)}} \]

      5. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}}} \]

      8. sqrt-unprod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

   8. Applied rewrites 44.2%

      \[\leadsto \sin th \cdot \color{blue}{\mathsf{copysign}\left(1, \sin ky\right)} \]

   if -0.70799999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

      3. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \]

   8. Applied rewrites 41.3%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{\sin ky}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{1}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \cdot \sin th} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2}}}{\sin ky}} \cdot \sin th} \]

   10. Applied rewrites 44.6%

       \[\leadsto \color{blue}{\frac{\sin ky}{\left|\sin kx\right|} \cdot \sin th} \]

   if 0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.02:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, \sin ky\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin th}{\frac{\sqrt{t\_1}}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (pow (sin kx) 2.0))
       (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0))))))
  (if (<= t_2 -0.02)
    (* (sin th) (copysign 1.0 (sin ky)))
    (if (<= t_2 5e-5) (/ (sin th) (/ (sqrt t_1) ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.02) {
		tmp = sin(th) * copysign(1.0, sin(ky));
	} else if (t_2 <= 5e-5) {
		tmp = sin(th) / (sqrt(t_1) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((t_1 + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.02) {
		tmp = Math.sin(th) * Math.copySign(1.0, Math.sin(ky));
	} else if (t_2 <= 5e-5) {
		tmp = Math.sin(th) / (Math.sqrt(t_1) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	t_2 = math.sin(ky) / math.sqrt((t_1 + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.02:
		tmp = math.sin(th) * math.copysign(1.0, math.sin(ky))
	elif t_2 <= 5e-5:
		tmp = math.sin(th) / (math.sqrt(t_1) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.02)
		tmp = Float64(sin(th) * copysign(1.0, sin(ky)));
	elseif (t_2 <= 5e-5)
		tmp = Float64(sin(th) / Float64(sqrt(t_1) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	t_2 = sin(ky) / sqrt((t_1 + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.02)
		tmp = sin(th) * (sign(sin(ky)) * abs(1.0));
	elseif (t_2 <= 5e-5)
		tmp = sin(th) / (sqrt(t_1) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.02], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[N[Sin[ky], $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-5], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] / 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))))) < -0.02

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      6. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}}} \]

      7. mult-flip N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\color{blue}{\frac{1}{2}}}} \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      17. lower-unsound-sqrt.f64 30.6%

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{0.5}} \]

   6. Applied rewrites 30.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{0.5}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{1 - \cos \left(ky + ky\right)}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{1 - \cos \left(ky + ky\right)}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \cos \left(ky + ky\right)}} \]

      5. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}}} \]

      8. sqrt-unprod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

   8. Applied rewrites 44.2%

      \[\leadsto \sin th \cdot \color{blue}{\mathsf{copysign}\left(1, \sin ky\right)} \]

   if -0.02 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000002e-5

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7%

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      7. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2}}}{ky}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{\color{blue}{ky}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{ky}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{ky}} \]

      4. lower-sin.f64 36.3%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin kx}^{2}}}{ky}} \]

   8. Applied rewrites 36.3%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2}}}{ky}}} \]

   if 5.0000000000000002e-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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

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 -0.02)
    (* (sin th) (copysign 1.0 (sin ky)))
    (if (<= t_1 5e-5) (* (sin th) (/ ky (fabs (sin kx)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = sin(th) * copysign(1.0, sin(ky));
	} else if (t_1 <= 5e-5) {
		tmp = sin(th) * (ky / fabs(sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = Math.sin(th) * Math.copySign(1.0, Math.sin(ky));
	} else if (t_1 <= 5e-5) {
		tmp = Math.sin(th) * (ky / Math.abs(Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.02:
		tmp = math.sin(th) * math.copysign(1.0, math.sin(ky))
	elif t_1 <= 5e-5:
		tmp = math.sin(th) * (ky / math.fabs(math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(sin(th) * copysign(1.0, sin(ky)));
	elseif (t_1 <= 5e-5)
		tmp = Float64(sin(th) * Float64(ky / abs(sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.02)
		tmp = sin(th) * (sign(sin(ky)) * abs(1.0));
	elseif (t_1 <= 5e-5)
		tmp = sin(th) * (ky / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[N[Sin[ky], $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-5], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.02

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      6. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}}} \]

      7. mult-flip N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\color{blue}{\frac{1}{2}}}} \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      17. lower-unsound-sqrt.f64 30.6%

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{0.5}} \]

   6. Applied rewrites 30.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{0.5}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{1 - \cos \left(ky + ky\right)}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{1 - \cos \left(ky + ky\right)}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \cos \left(ky + ky\right)}} \]

      5. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}}} \]

      8. sqrt-unprod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

   8. Applied rewrites 44.2%

      \[\leadsto \sin th \cdot \color{blue}{\mathsf{copysign}\left(1, \sin ky\right)} \]

   if -0.02 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000002e-5

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.3%

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. mult-flip N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \color{blue}{\frac{1}{\sqrt{{\sin kx}^{2}}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\sqrt{{\sin kx}^{2}}} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin th \cdot ky\right) \cdot \frac{\color{blue}{1}}{\sqrt{{\sin kx}^{2}}} \]

      5. associate-*l* N/A

         \[\leadsto \sin th \cdot \color{blue}{\left(ky \cdot \frac{1}{\sqrt{{\sin kx}^{2}}}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\left(ky \cdot \frac{1}{\sqrt{{\sin kx}^{2}}}\right)} \]

      7. mult-flip-rev N/A

         \[\leadsto \sin th \cdot \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      8. lower-/.f64 36.3%

         \[\leadsto \sin th \cdot \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \sin th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \sin th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \sin th \cdot \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \sin th \cdot \frac{ky}{\left|\sin kx\right|} \]

      13. lower-fabs.f64 39.5%

          \[\leadsto \sin th \cdot \frac{ky}{\left|\sin kx\right|} \]

   6. Applied rewrites 39.5%

      \[\leadsto \sin th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

   if 5.0000000000000002e-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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 56.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-262}:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right) \cdot \left|ky\right|}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_2 2e-262)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_2 2e-6)
       (/
        (*
         (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0))))
         (fabs ky))
        (fabs (sin kx)))
       (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= 2e-262) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_2 <= 2e-6) {
		tmp = ((th * (1.0 + (-0.16666666666666666 * pow(th, 2.0)))) * fabs(ky)) / fabs(sin(kx));
	} else {
		tmp = sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= 2e-262) {
		tmp = Math.sin(th) * Math.copySign(1.0, t_1);
	} else if (t_2 <= 2e-6) {
		tmp = ((th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0)))) * Math.abs(ky)) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_2 <= 2e-262:
		tmp = math.sin(th) * math.copysign(1.0, t_1)
	elif t_2 <= 2e-6:
		tmp = ((th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0)))) * math.fabs(ky)) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 2e-262)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_2 <= 2e-6)
		tmp = Float64(Float64(Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))) * abs(ky)) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= 2e-262)
		tmp = sin(th) * (sign(t_1) * abs(1.0));
	elseif (t_2 <= 2e-6)
		tmp = ((th * (1.0 + (-0.16666666666666666 * (th ^ 2.0)))) * abs(ky)) / abs(sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, 2e-262], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-6], N[(N[(N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-262

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \sin ky}} \]

      6. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}}} \]

      7. mult-flip N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

      9. sqrt-prod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      10. lower-unsound-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      11. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\color{blue}{\frac{1}{2}}}} \]

      12. cos-diff N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right) - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{\frac{1}{2}}} \]

      17. lower-unsound-sqrt.f64 30.6%

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{0.5}} \]

   6. Applied rewrites 30.6%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{0.5}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \color{blue}{\sqrt{\frac{1}{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{1 - \cos \left(ky + ky\right)}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{1 - \cos \left(ky + ky\right)}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \cos \left(ky + ky\right)}} \]

      5. sqrt-fabs-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \left|\sqrt{1 - \cos \left(ky + ky\right)}\right|} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2}} \cdot \sqrt{\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}}} \]

      8. sqrt-unprod N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(\sqrt{1 - \cos \left(ky + ky\right)} \cdot \sqrt{1 - \cos \left(ky + ky\right)}\right)}} \]

      11. rem-square-sqrt N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\frac{1}{2} \cdot \left(1 - \cos \left(ky + ky\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(1 - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}}} \]

   8. Applied rewrites 44.2%

      \[\leadsto \sin th \cdot \color{blue}{\mathsf{copysign}\left(1, \sin ky\right)} \]

   if 2e-262 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-6

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.3%

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-*.f64 35.3%

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\sin kx}^{2}}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\sin kx}^{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      8. lower-fabs.f64 37.6%

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

   6. Applied rewrites 37.6%

      \[\leadsto \frac{\sin th \cdot ky}{\color{blue}{\left|\sin kx\right|}} \]

   7. Taylor expanded in th around 0

      \[\leadsto \frac{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin kx\right|} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin kx\right|} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin kx\right|} \]

      4. lower-pow.f64 19.2%

         \[\leadsto \frac{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin kx\right|} \]

   9. Applied rewrites 19.2%

      \[\leadsto \frac{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]

   if 1.9999999999999999e-6 < (/.f64 (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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 44.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right) \cdot \left|ky\right|}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky))))
  (*
   (copysign 1.0 ky)
   (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 2e-6)
     (/
      (*
       (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0))))
       (fabs ky))
      (fabs (sin kx)))
     (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 2e-6) {
		tmp = ((th * (1.0 + (-0.16666666666666666 * pow(th, 2.0)))) * fabs(ky)) / fabs(sin(kx));
	} else {
		tmp = sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 2e-6) {
		tmp = ((th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0)))) * Math.abs(ky)) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 2e-6:
		tmp = ((th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0)))) * math.fabs(ky)) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 2e-6)
		tmp = Float64(Float64(Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))) * abs(ky)) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 2e-6)
		tmp = ((th * (1.0 + (-0.16666666666666666 * (th ^ 2.0)))) * abs(ky)) / abs(sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-6

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.3%

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-*.f64 35.3%

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\sin kx}^{2}}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\sin kx}^{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      8. lower-fabs.f64 37.6%

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

   6. Applied rewrites 37.6%

      \[\leadsto \frac{\sin th \cdot ky}{\color{blue}{\left|\sin kx\right|}} \]

   7. Taylor expanded in th around 0

      \[\leadsto \frac{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin kx\right|} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin kx\right|} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin kx\right|} \]

      4. lower-pow.f64 19.2%

         \[\leadsto \frac{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin kx\right|} \]

   9. Applied rewrites 19.2%

      \[\leadsto \frac{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right) \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]

   if 1.9999999999999999e-6 < (/.f64 (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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left|ky\right| \cdot th}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky))))
  (*
   (copysign 1.0 ky)
   (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 2e-6)
     (/ (* (fabs ky) th) (fabs (sin kx)))
     (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 2e-6) {
		tmp = (fabs(ky) * th) / fabs(sin(kx));
	} else {
		tmp = sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 2e-6) {
		tmp = (Math.abs(ky) * th) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 2e-6:
		tmp = (math.fabs(ky) * th) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 2e-6)
		tmp = Float64(Float64(abs(ky) * th) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 2e-6)
		tmp = (abs(ky) * th) / abs(sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[Abs[ky], $MachinePrecision] * th), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-6

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.3%

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-*.f64 35.3%

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\sin kx}^{2}}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\sin kx}^{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      8. lower-fabs.f64 37.6%

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

   6. Applied rewrites 37.6%

      \[\leadsto \frac{\sin th \cdot ky}{\color{blue}{\left|\sin kx\right|}} \]

   7. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\left|\color{blue}{\sin kx}\right|} \]
   8. Step-by-step derivation

      1. lower-*.f64 19.4%

         \[\leadsto \frac{ky \cdot th}{\left|\sin kx\right|} \]

   9. Applied rewrites 19.4%

      \[\leadsto \frac{ky \cdot th}{\left|\color{blue}{\sin kx}\right|} \]

   if 1.9999999999999999e-6 < (/.f64 (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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 43.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin \left(\left|kx\right|\right)}^{2} + {t\_1}^{2}}} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left|ky\right| \cdot \sin th}{\left|kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky))))
  (*
   (copysign 1.0 ky)
   (if (<=
        (/ t_1 (sqrt (+ (pow (sin (fabs kx)) 2.0) (pow t_1 2.0))))
        2e-12)
     (/ (* (fabs ky) (sin th)) (fabs kx))
     (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(fabs(kx)), 2.0) + pow(t_1, 2.0)))) <= 2e-12) {
		tmp = (fabs(ky) * sin(th)) / fabs(kx);
	} else {
		tmp = sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(Math.abs(kx)), 2.0) + Math.pow(t_1, 2.0)))) <= 2e-12) {
		tmp = (Math.abs(ky) * Math.sin(th)) / Math.abs(kx);
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(math.fabs(kx)), 2.0) + math.pow(t_1, 2.0)))) <= 2e-12:
		tmp = (math.fabs(ky) * math.sin(th)) / math.fabs(kx)
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(abs(kx)) ^ 2.0) + (t_1 ^ 2.0)))) <= 2e-12)
		tmp = Float64(Float64(abs(ky) * sin(th)) / abs(kx));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(abs(kx)) ^ 2.0) + (t_1 ^ 2.0)))) <= 2e-12)
		tmp = (abs(ky) * sin(th)) / abs(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-12], N[(N[(N[Abs[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-12

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.3%

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{kx} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{kx} \]

      3. lower-sin.f64 16.1%

         \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   7. Applied rewrites 16.1%

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]

   if 2e-12 < (/.f64 (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 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 36.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 4.5 \cdot 10^{-61}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky))))
  (*
   (copysign 1.0 ky)
   (if (<=
        (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))
        4.5e-61)
     (* -0.16666666666666666 (pow th 3.0))
     (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 4.5e-61) {
		tmp = -0.16666666666666666 * pow(th, 3.0);
	} else {
		tmp = sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 4.5e-61) {
		tmp = -0.16666666666666666 * Math.pow(th, 3.0);
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 4.5e-61:
		tmp = -0.16666666666666666 * math.pow(th, 3.0)
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 4.5e-61)
		tmp = Float64(-0.16666666666666666 * (th ^ 3.0));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 4.5e-61)
		tmp = -0.16666666666666666 * (th ^ 3.0);
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4.5e-61], N[(-0.16666666666666666 * N[Power[th, 3.0], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]), $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))))) < 4.5e-61

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{\color{blue}{2}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \]

      4. lower-pow.f64 12.7%

         \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

   10. Applied rewrites 12.7%

       \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]

   11. Taylor expanded in th around inf

       \[\leadsto \frac{-1}{6} \cdot {th}^{3} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot {th}^{3} \]

       2. lower-pow.f64 11.3%

          \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   13. Applied rewrites 11.3%

       \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   if 4.5e-61 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 22.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 4.5 \cdot 10^{-61}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;th \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky))))
  (*
   (copysign 1.0 ky)
   (if (<=
        (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))
        4.5e-61)
     (* -0.16666666666666666 (pow th 3.0))
     (* th (fma (* -0.16666666666666666 th) th 1.0))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 4.5e-61) {
		tmp = -0.16666666666666666 * pow(th, 3.0);
	} else {
		tmp = th * fma((-0.16666666666666666 * th), th, 1.0);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 4.5e-61)
		tmp = Float64(-0.16666666666666666 * (th ^ 3.0));
	else
		tmp = Float64(th * fma(Float64(-0.16666666666666666 * th), th, 1.0));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4.5e-61], N[(-0.16666666666666666 * N[Power[th, 3.0], $MachinePrecision]), $MachinePrecision], N[(th * N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th + 1.0), $MachinePrecision]), $MachinePrecision]]), $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))))) < 4.5e-61

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{\color{blue}{2}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \]

      4. lower-pow.f64 12.7%

         \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

   10. Applied rewrites 12.7%

       \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]

   11. Taylor expanded in th around inf

       \[\leadsto \frac{-1}{6} \cdot {th}^{3} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot {th}^{3} \]

       2. lower-pow.f64 11.3%

          \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   13. Applied rewrites 11.3%

       \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   if 4.5e-61 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.4%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.0%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.0%

      \[\leadsto \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{\color{blue}{2}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \]

      4. lower-pow.f64 12.7%

         \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

   10. Applied rewrites 12.7%

       \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{\color{blue}{2}}\right) \]

       2. +-commutative N/A

          \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \]

       3. lift-*.f64 N/A

          \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \]

       4. lift-pow.f64 N/A

          \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \]

       5. unpow2 N/A

          \[\leadsto th \cdot \left(\frac{-1}{6} \cdot \left(th \cdot th\right) + 1\right) \]

       6. associate-*r* N/A

          \[\leadsto th \cdot \left(\left(\frac{-1}{6} \cdot th\right) \cdot th + 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot th, th, 1\right) \]

       8. lower-*.f64 12.7%

          \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \]

   12. Applied rewrites 12.7%

       \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 15.9% accurate, 10.3× speedup

Math

\[\mathsf{copysign}\left(1, ky\right) \cdot \left(th \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right)\right) \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (* (copysign 1.0 ky) (* th (fma (* -0.16666666666666666 th) th 1.0))))

C

double code(double kx, double ky, double th) {
	return copysign(1.0, ky) * (th * fma((-0.16666666666666666 * th), th, 1.0));
}

Julia

function code(kx, ky, th)
	return Float64(copysign(1.0, ky) * Float64(th * fma(Float64(-0.16666666666666666 * th), th, 1.0)))
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(th * N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   7. lower-sin.f64 41.4%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

4. Applied rewrites 41.4%

   \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

5. Taylor expanded in ky around 0

   \[\leadsto \sin th \]
6. Step-by-step derivation

   1. lower-sin.f64 23.0%

      \[\leadsto \sin th \]

7. Applied rewrites 23.0%

   \[\leadsto \sin th \]

8. Taylor expanded in th around 0

   \[\leadsto th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right) \]

   2. lower-+.f64 N/A

      \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{\color{blue}{2}}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \]

   4. lower-pow.f64 12.7%

      \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

10. Applied rewrites 12.7%

    \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]
11. Step-by-step derivation

    1. lift-+.f64 N/A

       \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{\color{blue}{2}}\right) \]

    2. +-commutative N/A

       \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \]

    3. lift-*.f64 N/A

       \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \]

    4. lift-pow.f64 N/A

       \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \]

    5. unpow2 N/A

       \[\leadsto th \cdot \left(\frac{-1}{6} \cdot \left(th \cdot th\right) + 1\right) \]

    6. associate-*r* N/A

       \[\leadsto th \cdot \left(\left(\frac{-1}{6} \cdot th\right) \cdot th + 1\right) \]

    7. lower-fma.f64 N/A

       \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot th, th, 1\right) \]

    8. lower-*.f64 12.7%

       \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \]

12. Applied rewrites 12.7%

    \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025214 
(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)))