Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 6.3s

Alternatives: 19

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: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := {t\_1}^{2}\\ t_4 := \sqrt{t\_2 + t\_3}\\ t_5 := \frac{t\_1}{t\_4}\\ t_6 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -0.9999999998:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_5 \leq -0.1:\\ \;\;\;\;\frac{th \cdot t\_1}{t\_4}\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.995:\\ \;\;\;\;\frac{th}{t\_4} \cdot t\_1\\ \mathbf{elif}\;t\_5 \leq 1:\\ \;\;\;\;\frac{t\_1}{\sqrt{{kx}^{2} + t\_3}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, t\_6\right)} \cdot t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (pow (sin kx) 2.0))
       (t_3 (pow t_1 2.0))
       (t_4 (sqrt (+ t_2 t_3)))
       (t_5 (/ t_1 t_4))
       (t_6
        (*
         (fabs ky)
         (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -0.9999999998)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_5 -0.1)
       (/ (* th t_1) t_4)
       (if (<= t_5 5e-8)
         (* (/ t_1 (sqrt t_2)) (sin th))
         (if (<= t_5 0.995)
           (* (/ th t_4) t_1)
           (if (<= t_5 1.0)
             (* (/ t_1 (sqrt (+ (pow kx 2.0) t_3))) (sin th))
             (* (/ (sin th) (hypot (sin kx) t_6)) t_6)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(t_1, 2.0);
	double t_4 = sqrt((t_2 + t_3));
	double t_5 = t_1 / t_4;
	double t_6 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double tmp;
	if (t_5 <= -0.9999999998) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_5 <= -0.1) {
		tmp = (th * t_1) / t_4;
	} else if (t_5 <= 5e-8) {
		tmp = (t_1 / sqrt(t_2)) * sin(th);
	} else if (t_5 <= 0.995) {
		tmp = (th / t_4) * t_1;
	} else if (t_5 <= 1.0) {
		tmp = (t_1 / sqrt((pow(kx, 2.0) + t_3))) * sin(th);
	} else {
		tmp = (sin(th) / hypot(sin(kx), t_6)) * t_6;
	}
	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 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.pow(t_1, 2.0);
	double t_4 = Math.sqrt((t_2 + t_3));
	double t_5 = t_1 / t_4;
	double t_6 = Math.abs(ky) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(ky), 2.0)));
	double tmp;
	if (t_5 <= -0.9999999998) {
		tmp = Math.sin(th) * Math.copySign(1.0, t_1);
	} else if (t_5 <= -0.1) {
		tmp = (th * t_1) / t_4;
	} else if (t_5 <= 5e-8) {
		tmp = (t_1 / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_5 <= 0.995) {
		tmp = (th / t_4) * t_1;
	} else if (t_5 <= 1.0) {
		tmp = (t_1 / Math.sqrt((Math.pow(kx, 2.0) + t_3))) * Math.sin(th);
	} else {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), t_6)) * t_6;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.pow(t_1, 2.0)
	t_4 = math.sqrt((t_2 + t_3))
	t_5 = t_1 / t_4
	t_6 = math.fabs(ky) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(ky), 2.0)))
	tmp = 0
	if t_5 <= -0.9999999998:
		tmp = math.sin(th) * math.copysign(1.0, t_1)
	elif t_5 <= -0.1:
		tmp = (th * t_1) / t_4
	elif t_5 <= 5e-8:
		tmp = (t_1 / math.sqrt(t_2)) * math.sin(th)
	elif t_5 <= 0.995:
		tmp = (th / t_4) * t_1
	elif t_5 <= 1.0:
		tmp = (t_1 / math.sqrt((math.pow(kx, 2.0) + t_3))) * math.sin(th)
	else:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), t_6)) * t_6
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(kx) ^ 2.0
	t_3 = t_1 ^ 2.0
	t_4 = sqrt(Float64(t_2 + t_3))
	t_5 = Float64(t_1 / t_4)
	t_6 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	tmp = 0.0
	if (t_5 <= -0.9999999998)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_5 <= -0.1)
		tmp = Float64(Float64(th * t_1) / t_4);
	elseif (t_5 <= 5e-8)
		tmp = Float64(Float64(t_1 / sqrt(t_2)) * sin(th));
	elseif (t_5 <= 0.995)
		tmp = Float64(Float64(th / t_4) * t_1);
	elseif (t_5 <= 1.0)
		tmp = Float64(Float64(t_1 / sqrt(Float64((kx ^ 2.0) + t_3))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), t_6)) * t_6);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = sin(kx) ^ 2.0;
	t_3 = t_1 ^ 2.0;
	t_4 = sqrt((t_2 + t_3));
	t_5 = t_1 / t_4;
	t_6 = abs(ky) * (1.0 + (-0.16666666666666666 * (abs(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_5 <= -0.9999999998)
		tmp = sin(th) * (sign(t_1) * abs(1.0));
	elseif (t_5 <= -0.1)
		tmp = (th * t_1) / t_4;
	elseif (t_5 <= 5e-8)
		tmp = (t_1 / sqrt(t_2)) * sin(th);
	elseif (t_5 <= 0.995)
		tmp = (th / t_4) * t_1;
	elseif (t_5 <= 1.0)
		tmp = (t_1 / sqrt(((kx ^ 2.0) + t_3))) * sin(th);
	else
		tmp = (sin(th) / hypot(sin(kx), t_6)) * t_6;
	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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 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$5, -0.9999999998], 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$5, -0.1], N[(N[(th * t$95$1), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 5e-8], N[(N[(t$95$1 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.995], N[(N[(th / t$95$4), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$5, 1.0], N[(N[(t$95$1 / N[Sqrt[N[(N[Power[kx, 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$6 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$6), $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))))) < -0.99999999979999998

   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-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.99999999979999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 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 45.8%

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

   4. Applied rewrites 45.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   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

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.2%

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

   4. Applied rewrites 41.2%

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

   if 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.995

   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 47.7%

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

   6. Applied rewrites 47.7%

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

   if 0.995 < (/.f64 (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

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

      1. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

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

   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 ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5%

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

   6. Applied rewrites 52.5%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.7%

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

   9. Applied rewrites 54.7%

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

Alternative 3: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := {t\_2}^{2}\\ t_4 := {\sin kx}^{2}\\ t_5 := \sqrt{t\_4 + t\_3}\\ t_6 := \frac{t\_2}{t\_5}\\ t_7 := \frac{th}{t\_5} \cdot t\_2\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_6 \leq -0.9999999998:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_2\right)\\ \mathbf{elif}\;t\_6 \leq -0.1:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_4}} \cdot \sin th\\ \mathbf{elif}\;t\_6 \leq 0.995:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_6 \leq 1:\\ \;\;\;\;\frac{t\_2}{\sqrt{{kx}^{2} + t\_3}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \end{array} \end{array} \]

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 (pow (sin kx) 2.0))
       (t_5 (sqrt (+ t_4 t_3)))
       (t_6 (/ t_2 t_5))
       (t_7 (* (/ th t_5) t_2)))
  (*
   (copysign 1.0 ky)
   (if (<= t_6 -0.9999999998)
     (* (sin th) (copysign 1.0 t_2))
     (if (<= t_6 -0.1)
       t_7
       (if (<= t_6 5e-8)
         (* (/ t_2 (sqrt t_4)) (sin th))
         (if (<= t_6 0.995)
           t_7
           (if (<= t_6 1.0)
             (* (/ t_2 (sqrt (+ (pow kx 2.0) t_3))) (sin th))
             (* (/ (sin th) (hypot (sin kx) t_1)) t_1)))))))))

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 = pow(sin(kx), 2.0);
	double t_5 = sqrt((t_4 + t_3));
	double t_6 = t_2 / t_5;
	double t_7 = (th / t_5) * t_2;
	double tmp;
	if (t_6 <= -0.9999999998) {
		tmp = sin(th) * copysign(1.0, t_2);
	} else if (t_6 <= -0.1) {
		tmp = t_7;
	} else if (t_6 <= 5e-8) {
		tmp = (t_2 / sqrt(t_4)) * sin(th);
	} else if (t_6 <= 0.995) {
		tmp = t_7;
	} else if (t_6 <= 1.0) {
		tmp = (t_2 / sqrt((pow(kx, 2.0) + t_3))) * sin(th);
	} else {
		tmp = (sin(th) / hypot(sin(kx), t_1)) * t_1;
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(ky) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(ky), 2.0)));
	double t_2 = Math.sin(Math.abs(ky));
	double t_3 = Math.pow(t_2, 2.0);
	double t_4 = Math.pow(Math.sin(kx), 2.0);
	double t_5 = Math.sqrt((t_4 + t_3));
	double t_6 = t_2 / t_5;
	double t_7 = (th / t_5) * t_2;
	double tmp;
	if (t_6 <= -0.9999999998) {
		tmp = Math.sin(th) * Math.copySign(1.0, t_2);
	} else if (t_6 <= -0.1) {
		tmp = t_7;
	} else if (t_6 <= 5e-8) {
		tmp = (t_2 / Math.sqrt(t_4)) * Math.sin(th);
	} else if (t_6 <= 0.995) {
		tmp = t_7;
	} else if (t_6 <= 1.0) {
		tmp = (t_2 / Math.sqrt((Math.pow(kx, 2.0) + t_3))) * Math.sin(th);
	} else {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), t_1)) * t_1;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(ky) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(ky), 2.0)))
	t_2 = math.sin(math.fabs(ky))
	t_3 = math.pow(t_2, 2.0)
	t_4 = math.pow(math.sin(kx), 2.0)
	t_5 = math.sqrt((t_4 + t_3))
	t_6 = t_2 / t_5
	t_7 = (th / t_5) * t_2
	tmp = 0
	if t_6 <= -0.9999999998:
		tmp = math.sin(th) * math.copysign(1.0, t_2)
	elif t_6 <= -0.1:
		tmp = t_7
	elif t_6 <= 5e-8:
		tmp = (t_2 / math.sqrt(t_4)) * math.sin(th)
	elif t_6 <= 0.995:
		tmp = t_7
	elif t_6 <= 1.0:
		tmp = (t_2 / math.sqrt((math.pow(kx, 2.0) + t_3))) * math.sin(th)
	else:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), t_1)) * t_1
	return math.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 = sin(kx) ^ 2.0
	t_5 = sqrt(Float64(t_4 + t_3))
	t_6 = Float64(t_2 / t_5)
	t_7 = Float64(Float64(th / t_5) * t_2)
	tmp = 0.0
	if (t_6 <= -0.9999999998)
		tmp = Float64(sin(th) * copysign(1.0, t_2));
	elseif (t_6 <= -0.1)
		tmp = t_7;
	elseif (t_6 <= 5e-8)
		tmp = Float64(Float64(t_2 / sqrt(t_4)) * sin(th));
	elseif (t_6 <= 0.995)
		tmp = t_7;
	elseif (t_6 <= 1.0)
		tmp = Float64(Float64(t_2 / sqrt(Float64((kx ^ 2.0) + t_3))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), t_1)) * t_1);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(ky) * (1.0 + (-0.16666666666666666 * (abs(ky) ^ 2.0)));
	t_2 = sin(abs(ky));
	t_3 = t_2 ^ 2.0;
	t_4 = sin(kx) ^ 2.0;
	t_5 = sqrt((t_4 + t_3));
	t_6 = t_2 / t_5;
	t_7 = (th / t_5) * t_2;
	tmp = 0.0;
	if (t_6 <= -0.9999999998)
		tmp = sin(th) * (sign(t_2) * abs(1.0));
	elseif (t_6 <= -0.1)
		tmp = t_7;
	elseif (t_6 <= 5e-8)
		tmp = (t_2 / sqrt(t_4)) * sin(th);
	elseif (t_6 <= 0.995)
		tmp = t_7;
	elseif (t_6 <= 1.0)
		tmp = (t_2 / sqrt(((kx ^ 2.0) + t_3))) * sin(th);
	else
		tmp = (sin(th) / hypot(sin(kx), t_1)) * t_1;
	end
	tmp_2 = (sign(ky) * abs(1.0)) * 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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(t$95$4 + t$95$3), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[(th / t$95$5), $MachinePrecision] * 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$6, -0.9999999998], 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$6, -0.1], t$95$7, If[LessEqual[t$95$6, 5e-8], N[(N[(t$95$2 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 0.995], t$95$7, If[LessEqual[t$95$6, 1.0], 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[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $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))))) < -0.99999999979999998

   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-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.99999999979999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.995

   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 47.7%

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

   6. Applied rewrites 47.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   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

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.2%

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

   4. Applied rewrites 41.2%

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

   if 0.995 < (/.f64 (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

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

      1. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

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

   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 ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5%

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

   6. Applied rewrites 52.5%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.7%

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

   9. Applied rewrites 54.7%

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

Alternative 4: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := \sqrt{t\_2 + {t\_1}^{2}}\\ t_4 := \frac{t\_1}{t\_3}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.9999999998:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;\frac{th \cdot t\_1}{t\_3}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;\frac{th}{t\_3} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(kx \cdot \left(1 + -0.16666666666666666 \cdot {kx}^{2}\right), t\_1\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (pow (sin kx) 2.0))
       (t_3 (sqrt (+ t_2 (pow t_1 2.0))))
       (t_4 (/ t_1 t_3)))
  (*
   (copysign 1.0 ky)
   (if (<= t_4 -0.9999999998)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_4 -0.1)
       (/ (* th t_1) t_3)
       (if (<= t_4 5e-8)
         (* (/ t_1 (sqrt t_2)) (sin th))
         (if (<= t_4 0.995)
           (* (/ th t_3) t_1)
           (*
            (/
             (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 = pow(sin(kx), 2.0);
	double t_3 = sqrt((t_2 + pow(t_1, 2.0)));
	double t_4 = t_1 / t_3;
	double tmp;
	if (t_4 <= -0.9999999998) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_4 <= -0.1) {
		tmp = (th * t_1) / t_3;
	} else if (t_4 <= 5e-8) {
		tmp = (t_1 / sqrt(t_2)) * sin(th);
	} else if (t_4 <= 0.995) {
		tmp = (th / t_3) * t_1;
	} else {
		tmp = (sin(th) / hypot((kx * (1.0 + (-0.16666666666666666 * pow(kx, 2.0)))), t_1)) * 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 t_2 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.sqrt((t_2 + Math.pow(t_1, 2.0)));
	double t_4 = t_1 / t_3;
	double tmp;
	if (t_4 <= -0.9999999998) {
		tmp = Math.sin(th) * Math.copySign(1.0, t_1);
	} else if (t_4 <= -0.1) {
		tmp = (th * t_1) / t_3;
	} else if (t_4 <= 5e-8) {
		tmp = (t_1 / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_4 <= 0.995) {
		tmp = (th / t_3) * t_1;
	} else {
		tmp = (Math.sin(th) / Math.hypot((kx * (1.0 + (-0.16666666666666666 * Math.pow(kx, 2.0)))), t_1)) * t_1;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.sqrt((t_2 + math.pow(t_1, 2.0)))
	t_4 = t_1 / t_3
	tmp = 0
	if t_4 <= -0.9999999998:
		tmp = math.sin(th) * math.copysign(1.0, t_1)
	elif t_4 <= -0.1:
		tmp = (th * t_1) / t_3
	elif t_4 <= 5e-8:
		tmp = (t_1 / math.sqrt(t_2)) * math.sin(th)
	elif t_4 <= 0.995:
		tmp = (th / t_3) * t_1
	else:
		tmp = (math.sin(th) / math.hypot((kx * (1.0 + (-0.16666666666666666 * math.pow(kx, 2.0)))), t_1)) * t_1
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(kx) ^ 2.0
	t_3 = sqrt(Float64(t_2 + (t_1 ^ 2.0)))
	t_4 = Float64(t_1 / t_3)
	tmp = 0.0
	if (t_4 <= -0.9999999998)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_4 <= -0.1)
		tmp = Float64(Float64(th * t_1) / t_3);
	elseif (t_4 <= 5e-8)
		tmp = Float64(Float64(t_1 / sqrt(t_2)) * sin(th));
	elseif (t_4 <= 0.995)
		tmp = Float64(Float64(th / t_3) * t_1);
	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

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = sin(kx) ^ 2.0;
	t_3 = sqrt((t_2 + (t_1 ^ 2.0)));
	t_4 = t_1 / t_3;
	tmp = 0.0;
	if (t_4 <= -0.9999999998)
		tmp = sin(th) * (sign(t_1) * abs(1.0));
	elseif (t_4 <= -0.1)
		tmp = (th * t_1) / t_3;
	elseif (t_4 <= 5e-8)
		tmp = (t_1 / sqrt(t_2)) * sin(th);
	elseif (t_4 <= 0.995)
		tmp = (th / t_3) * t_1;
	else
		tmp = (sin(th) / hypot((kx * (1.0 + (-0.16666666666666666 * (kx ^ 2.0)))), t_1)) * t_1;
	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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / 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, -0.9999999998], 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$4, -0.1], N[(N[(th * t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 5e-8], N[(N[(t$95$1 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], N[(N[(th / t$95$3), $MachinePrecision] * t$95$1), $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))))) < -0.99999999979999998

   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-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.99999999979999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 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 45.8%

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

   4. Applied rewrites 45.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   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

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.2%

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

   4. Applied rewrites 41.2%

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

   if 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.995

   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 47.7%

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

   6. Applied rewrites 47.7%

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

   if 0.995 < (/.f64 (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 58.6%

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

   6. Applied rewrites 58.6%

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

Alternative 5: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := \sqrt{t\_2 + {t\_1}^{2}}\\ t_4 := \frac{t\_1}{t\_3}\\ t_5 := \frac{th}{t\_3} \cdot t\_1\\ t_6 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.9999999998:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9999999820132889:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, t\_6\right)} \cdot t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (pow (sin kx) 2.0))
       (t_3 (sqrt (+ t_2 (pow t_1 2.0))))
       (t_4 (/ t_1 t_3))
       (t_5 (* (/ th t_3) t_1))
       (t_6
        (*
         (fabs ky)
         (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_4 -0.9999999998)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_4 -0.1)
       t_5
       (if (<= t_4 5e-8)
         (* (/ t_1 (sqrt t_2)) (sin th))
         (if (<= t_4 0.9999999820132889)
           t_5
           (if (<= t_4 1.0)
             (sin th)
             (* (/ (sin th) (hypot (sin kx) t_6)) t_6)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sqrt((t_2 + pow(t_1, 2.0)));
	double t_4 = t_1 / t_3;
	double t_5 = (th / t_3) * t_1;
	double t_6 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double tmp;
	if (t_4 <= -0.9999999998) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_4 <= -0.1) {
		tmp = t_5;
	} else if (t_4 <= 5e-8) {
		tmp = (t_1 / sqrt(t_2)) * sin(th);
	} else if (t_4 <= 0.9999999820132889) {
		tmp = t_5;
	} else if (t_4 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) / hypot(sin(kx), t_6)) * t_6;
	}
	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 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.sqrt((t_2 + Math.pow(t_1, 2.0)));
	double t_4 = t_1 / t_3;
	double t_5 = (th / t_3) * t_1;
	double t_6 = Math.abs(ky) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(ky), 2.0)));
	double tmp;
	if (t_4 <= -0.9999999998) {
		tmp = Math.sin(th) * Math.copySign(1.0, t_1);
	} else if (t_4 <= -0.1) {
		tmp = t_5;
	} else if (t_4 <= 5e-8) {
		tmp = (t_1 / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_4 <= 0.9999999820132889) {
		tmp = t_5;
	} else if (t_4 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), t_6)) * t_6;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.sqrt((t_2 + math.pow(t_1, 2.0)))
	t_4 = t_1 / t_3
	t_5 = (th / t_3) * t_1
	t_6 = math.fabs(ky) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(ky), 2.0)))
	tmp = 0
	if t_4 <= -0.9999999998:
		tmp = math.sin(th) * math.copysign(1.0, t_1)
	elif t_4 <= -0.1:
		tmp = t_5
	elif t_4 <= 5e-8:
		tmp = (t_1 / math.sqrt(t_2)) * math.sin(th)
	elif t_4 <= 0.9999999820132889:
		tmp = t_5
	elif t_4 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), t_6)) * t_6
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(kx) ^ 2.0
	t_3 = sqrt(Float64(t_2 + (t_1 ^ 2.0)))
	t_4 = Float64(t_1 / t_3)
	t_5 = Float64(Float64(th / t_3) * t_1)
	t_6 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.9999999998)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_4 <= -0.1)
		tmp = t_5;
	elseif (t_4 <= 5e-8)
		tmp = Float64(Float64(t_1 / sqrt(t_2)) * sin(th));
	elseif (t_4 <= 0.9999999820132889)
		tmp = t_5;
	elseif (t_4 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), t_6)) * t_6);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = sin(kx) ^ 2.0;
	t_3 = sqrt((t_2 + (t_1 ^ 2.0)));
	t_4 = t_1 / t_3;
	t_5 = (th / t_3) * t_1;
	t_6 = abs(ky) * (1.0 + (-0.16666666666666666 * (abs(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_4 <= -0.9999999998)
		tmp = sin(th) * (sign(t_1) * abs(1.0));
	elseif (t_4 <= -0.1)
		tmp = t_5;
	elseif (t_4 <= 5e-8)
		tmp = (t_1 / sqrt(t_2)) * sin(th);
	elseif (t_4 <= 0.9999999820132889)
		tmp = t_5;
	elseif (t_4 <= 1.0)
		tmp = sin(th);
	else
		tmp = (sin(th) / hypot(sin(kx), t_6)) * t_6;
	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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(th / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 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$4, -0.9999999998], 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$4, -0.1], t$95$5, If[LessEqual[t$95$4, 5e-8], N[(N[(t$95$1 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9999999820132889], t$95$5, If[LessEqual[t$95$4, 1.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$6 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99999999979999998

   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-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.99999999979999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999998201328888

   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 47.7%

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

   6. Applied rewrites 47.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   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

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.2%

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

   4. Applied rewrites 41.2%

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

   if 0.99999998201328888 < (/.f64 (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. Taylor expanded in ky around 0

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

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

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

   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 ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5%

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

   6. Applied rewrites 52.5%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.7%

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

   9. Applied rewrites 54.7%

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

Alternative 6: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \mathsf{hypot}\left(t\_2, \sin kx\right)\\ t_4 := {\sin kx}^{2}\\ t_5 := \frac{t\_2}{\sqrt{t\_4 + {t\_2}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -0.9999999998:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_2\right)\\ \mathbf{elif}\;t\_5 \leq -0.1:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot t\_2}{t\_3}\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_4}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.9999999820132889:\\ \;\;\;\;\frac{t\_2}{t\_3} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, -0.16666666666666666, th\right)\\ \mathbf{elif}\;t\_5 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \end{array} \end{array} \]

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 (hypot t_2 (sin kx)))
       (t_4 (pow (sin kx) 2.0))
       (t_5 (/ t_2 (sqrt (+ t_4 (pow t_2 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -0.9999999998)
     (* (sin th) (copysign 1.0 t_2))
     (if (<= t_5 -0.1)
       (/ (* (* (fma (* th th) -0.16666666666666666 1.0) th) t_2) t_3)
       (if (<= t_5 5e-8)
         (* (/ t_2 (sqrt t_4)) (sin th))
         (if (<= t_5 0.9999999820132889)
           (*
            (/ t_2 t_3)
            (fma (* (* th th) th) -0.16666666666666666 th))
           (if (<= t_5 1.0)
             (sin th)
             (* (/ (sin th) (hypot (sin kx) t_1)) t_1)))))))))

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 = hypot(t_2, sin(kx));
	double t_4 = pow(sin(kx), 2.0);
	double t_5 = t_2 / sqrt((t_4 + pow(t_2, 2.0)));
	double tmp;
	if (t_5 <= -0.9999999998) {
		tmp = sin(th) * copysign(1.0, t_2);
	} else if (t_5 <= -0.1) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * th) * t_2) / t_3;
	} else if (t_5 <= 5e-8) {
		tmp = (t_2 / sqrt(t_4)) * sin(th);
	} else if (t_5 <= 0.9999999820132889) {
		tmp = (t_2 / t_3) * fma(((th * th) * th), -0.16666666666666666, th);
	} else if (t_5 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) / hypot(sin(kx), t_1)) * t_1;
	}
	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 = hypot(t_2, sin(kx))
	t_4 = sin(kx) ^ 2.0
	t_5 = Float64(t_2 / sqrt(Float64(t_4 + (t_2 ^ 2.0))))
	tmp = 0.0
	if (t_5 <= -0.9999999998)
		tmp = Float64(sin(th) * copysign(1.0, t_2));
	elseif (t_5 <= -0.1)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * t_2) / t_3);
	elseif (t_5 <= 5e-8)
		tmp = Float64(Float64(t_2 / sqrt(t_4)) * sin(th));
	elseif (t_5 <= 0.9999999820132889)
		tmp = Float64(Float64(t_2 / t_3) * fma(Float64(Float64(th * th) * th), -0.16666666666666666, th));
	elseif (t_5 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), t_1)) * t_1);
	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[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / N[Sqrt[N[(t$95$4 + N[Power[t$95$2, 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$5, -0.9999999998], 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, -0.1], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$5, 5e-8], N[(N[(t$95$2 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.9999999820132889], N[(N[(t$95$2 / t$95$3), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666 + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $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))))) < -0.99999999979999998

   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-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.99999999979999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 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 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   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. lift-hypot.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-hypot.f64 N/A

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

      11. lower-/.f64 N/A

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

   8. Applied rewrites 46.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   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

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.2%

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

   4. Applied rewrites 41.2%

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

   if 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999998201328888

   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 50.4%

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

   6. Applied rewrites 50.4%

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

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. cube-unmult N/A

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

      14. metadata-eval N/A

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

      15. pow-plus N/A

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

      16. lift-pow.f64 N/A

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

      17. lower-*.f64 50.4%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 50.4%

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

   8. Applied rewrites 50.4%

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

   if 0.99999998201328888 < (/.f64 (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. Taylor expanded in ky around 0

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

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

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

   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 ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 52.5%

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

   6. Applied rewrites 52.5%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.7%

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

   9. Applied rewrites 54.7%

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

Alternative 7: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \mathsf{hypot}\left(t\_1, \sin kx\right)\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{t\_1}{\sqrt{t\_3 + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.9999999998:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot t\_1}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9999999820132889:\\ \;\;\;\;\frac{t\_1}{t\_2} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, -0.16666666666666666, th\right)\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (hypot t_1 (sin kx)))
       (t_3 (pow (sin kx) 2.0))
       (t_4 (/ t_1 (sqrt (+ t_3 (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_4 -0.9999999998)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_4 -0.1)
       (/ (* (* (fma (* th th) -0.16666666666666666 1.0) th) t_1) t_2)
       (if (<= t_4 5e-8)
         (* (/ t_1 (sqrt t_3)) (sin th))
         (if (<= t_4 0.9999999820132889)
           (*
            (/ t_1 t_2)
            (fma (* (* th th) th) -0.16666666666666666 th))
           (if (<= t_4 2.0)
             (sin th)
             (* (sin th) (/ (fabs ky) (fabs (sin kx))))))))))))

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 = pow(sin(kx), 2.0);
	double t_4 = t_1 / sqrt((t_3 + pow(t_1, 2.0)));
	double tmp;
	if (t_4 <= -0.9999999998) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_4 <= -0.1) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * th) * t_1) / t_2;
	} else if (t_4 <= 5e-8) {
		tmp = (t_1 / sqrt(t_3)) * sin(th);
	} else if (t_4 <= 0.9999999820132889) {
		tmp = (t_1 / t_2) * fma(((th * th) * th), -0.16666666666666666, th);
	} else if (t_4 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	}
	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 = sin(kx) ^ 2.0
	t_4 = Float64(t_1 / sqrt(Float64(t_3 + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.9999999998)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_4 <= -0.1)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * t_1) / t_2);
	elseif (t_4 <= 5e-8)
		tmp = Float64(Float64(t_1 / sqrt(t_3)) * sin(th));
	elseif (t_4 <= 0.9999999820132889)
		tmp = Float64(Float64(t_1 / t_2) * fma(Float64(Float64(th * th) * th), -0.16666666666666666, th));
	elseif (t_4 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[Sqrt[N[(t$95$3 + 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$4, -0.9999999998], 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$4, -0.1], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 5e-8], N[(N[(t$95$1 / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9999999820132889], N[(N[(t$95$1 / t$95$2), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666 + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $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))))) < -0.99999999979999998

   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-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.99999999979999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 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 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   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. lift-hypot.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-hypot.f64 N/A

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

      11. lower-/.f64 N/A

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

   8. Applied rewrites 46.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   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

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.2%

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

   4. Applied rewrites 41.2%

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

   if 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999998201328888

   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 50.4%

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

   6. Applied rewrites 50.4%

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

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. cube-unmult N/A

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

      14. metadata-eval N/A

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

      15. pow-plus N/A

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

      16. lift-pow.f64 N/A

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

      17. lower-*.f64 50.4%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 50.4%

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

   8. Applied rewrites 50.4%

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

   if 0.99999998201328888 < (/.f64 (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 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

   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. 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 35.0%

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

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   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 36.1%

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

      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 39.2%

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

   6. Applied rewrites 39.2%

      \[\leadsto \sin th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 8: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, -0.16666666666666666, th\right)\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{t\_1}{\sqrt{t\_3 + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.9999999998:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9999999820132889:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2
        (*
         (/ t_1 (hypot t_1 (sin kx)))
         (fma (* (* th th) th) -0.16666666666666666 th)))
       (t_3 (pow (sin kx) 2.0))
       (t_4 (/ t_1 (sqrt (+ t_3 (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_4 -0.9999999998)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_4 -0.1)
       t_2
       (if (<= t_4 5e-8)
         (* (/ t_1 (sqrt t_3)) (sin th))
         (if (<= t_4 0.9999999820132889)
           t_2
           (if (<= t_4 2.0)
             (sin th)
             (* (sin th) (/ (fabs ky) (fabs (sin kx))))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = (t_1 / hypot(t_1, sin(kx))) * fma(((th * th) * th), -0.16666666666666666, th);
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = t_1 / sqrt((t_3 + pow(t_1, 2.0)));
	double tmp;
	if (t_4 <= -0.9999999998) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_4 <= -0.1) {
		tmp = t_2;
	} else if (t_4 <= 5e-8) {
		tmp = (t_1 / sqrt(t_3)) * sin(th);
	} else if (t_4 <= 0.9999999820132889) {
		tmp = t_2;
	} else if (t_4 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * fma(Float64(Float64(th * th) * th), -0.16666666666666666, th))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(t_1 / sqrt(Float64(t_3 + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.9999999998)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_4 <= -0.1)
		tmp = t_2;
	elseif (t_4 <= 5e-8)
		tmp = Float64(Float64(t_1 / sqrt(t_3)) * sin(th));
	elseif (t_4 <= 0.9999999820132889)
		tmp = t_2;
	elseif (t_4 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	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[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666 + th), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[Sqrt[N[(t$95$3 + 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$4, -0.9999999998], 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$4, -0.1], t$95$2, If[LessEqual[t$95$4, 5e-8], N[(N[(t$95$1 / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9999999820132889], t$95$2, If[LessEqual[t$95$4, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $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))))) < -0.99999999979999998

   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-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.99999999979999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999998201328888

   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 50.4%

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

   6. Applied rewrites 50.4%

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

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. cube-unmult N/A

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

      14. metadata-eval N/A

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

      15. pow-plus N/A

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

      16. lift-pow.f64 N/A

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

      17. lower-*.f64 50.4%

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 50.4%

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

   8. Applied rewrites 50.4%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   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

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.2%

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

   4. Applied rewrites 41.2%

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

   if 0.99999998201328888 < (/.f64 (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 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

   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. 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 35.0%

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

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   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 36.1%

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

      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 39.2%

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

   6. Applied rewrites 39.2%

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

Alternative 9: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := t\_1 \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\mathsf{hypot}\left(t\_1, \sin kx\right)}\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{t\_1}{\sqrt{t\_3 + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.9999999998:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9999999820132889:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2
        (*
         t_1
         (/
          (* (fma (* th th) -0.16666666666666666 1.0) th)
          (hypot t_1 (sin kx)))))
       (t_3 (pow (sin kx) 2.0))
       (t_4 (/ t_1 (sqrt (+ t_3 (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_4 -0.9999999998)
     (* (sin th) (copysign 1.0 t_1))
     (if (<= t_4 -0.1)
       t_2
       (if (<= t_4 5e-8)
         (* (/ t_1 (sqrt t_3)) (sin th))
         (if (<= t_4 0.9999999820132889)
           t_2
           (if (<= t_4 2.0)
             (sin th)
             (* (sin th) (/ (fabs ky) (fabs (sin kx))))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 * ((fma((th * th), -0.16666666666666666, 1.0) * th) / hypot(t_1, sin(kx)));
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = t_1 / sqrt((t_3 + pow(t_1, 2.0)));
	double tmp;
	if (t_4 <= -0.9999999998) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_4 <= -0.1) {
		tmp = t_2;
	} else if (t_4 <= 5e-8) {
		tmp = (t_1 / sqrt(t_3)) * sin(th);
	} else if (t_4 <= 0.9999999820132889) {
		tmp = t_2;
	} else if (t_4 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(t_1 * Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / hypot(t_1, sin(kx))))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(t_1 / sqrt(Float64(t_3 + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.9999999998)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_4 <= -0.1)
		tmp = t_2;
	elseif (t_4 <= 5e-8)
		tmp = Float64(Float64(t_1 / sqrt(t_3)) * sin(th));
	elseif (t_4 <= 0.9999999820132889)
		tmp = t_2;
	elseif (t_4 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	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[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[Sqrt[N[(t$95$3 + 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$4, -0.9999999998], 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$4, -0.1], t$95$2, If[LessEqual[t$95$4, 5e-8], N[(N[(t$95$1 / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9999999820132889], t$95$2, If[LessEqual[t$95$4, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $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))))) < -0.99999999979999998

   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-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.99999999979999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99999998201328888

   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 50.4%

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

   6. Applied rewrites 50.4%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   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. lift-hypot.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-hypot.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 50.4%

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

   8. Applied rewrites 50.4%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8

   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

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.2%

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

   4. Applied rewrites 41.2%

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

   if 0.99999998201328888 < (/.f64 (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 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

   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. 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 35.0%

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

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   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 36.1%

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

      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 39.2%

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

   6. Applied rewrites 39.2%

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

Alternative 10: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{t\_1}{\sqrt{t\_3 + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.71:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.7:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (* (sin th) (copysign 1.0 t_1)))
       (t_3 (pow (sin kx) 2.0))
       (t_4 (/ t_1 (sqrt (+ t_3 (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_4 -0.71)
     t_2
     (if (<= t_4 0.7)
       (* (/ t_1 (sqrt t_3)) (sin th))
       (if (<= t_4 2.0)
         t_2
         (* (sin th) (/ (fabs ky) (fabs (sin kx))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(th) * copysign(1.0, t_1);
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = t_1 / sqrt((t_3 + pow(t_1, 2.0)));
	double tmp;
	if (t_4 <= -0.71) {
		tmp = t_2;
	} else if (t_4 <= 0.7) {
		tmp = (t_1 / sqrt(t_3)) * sin(th);
	} else if (t_4 <= 2.0) {
		tmp = t_2;
	} else {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	}
	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 = Math.sin(th) * Math.copySign(1.0, t_1);
	double t_3 = Math.pow(Math.sin(kx), 2.0);
	double t_4 = t_1 / Math.sqrt((t_3 + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_4 <= -0.71) {
		tmp = t_2;
	} else if (t_4 <= 0.7) {
		tmp = (t_1 / Math.sqrt(t_3)) * Math.sin(th);
	} else if (t_4 <= 2.0) {
		tmp = t_2;
	} else {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(Math.sin(kx)));
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.sin(th) * math.copysign(1.0, t_1)
	t_3 = math.pow(math.sin(kx), 2.0)
	t_4 = t_1 / math.sqrt((t_3 + math.pow(t_1, 2.0)))
	tmp = 0
	if t_4 <= -0.71:
		tmp = t_2
	elif t_4 <= 0.7:
		tmp = (t_1 / math.sqrt(t_3)) * math.sin(th)
	elif t_4 <= 2.0:
		tmp = t_2
	else:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(math.sin(kx)))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(sin(th) * copysign(1.0, t_1))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(t_1 / sqrt(Float64(t_3 + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.71)
		tmp = t_2;
	elseif (t_4 <= 0.7)
		tmp = Float64(Float64(t_1 / sqrt(t_3)) * sin(th));
	elseif (t_4 <= 2.0)
		tmp = t_2;
	else
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = sin(th) * (sign(t_1) * abs(1.0));
	t_3 = sin(kx) ^ 2.0;
	t_4 = t_1 / sqrt((t_3 + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_4 <= -0.71)
		tmp = t_2;
	elseif (t_4 <= 0.7)
		tmp = (t_1 / sqrt(t_3)) * sin(th);
	elseif (t_4 <= 2.0)
		tmp = t_2;
	else
		tmp = sin(th) * (abs(ky) / abs(sin(kx)));
	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[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[Sqrt[N[(t$95$3 + 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$4, -0.71], t$95$2, If[LessEqual[t$95$4, 0.7], N[(N[(t$95$1 / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], t$95$2, N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.70999999999999996 or 0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 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-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.70999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   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

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.2%

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

   4. Applied rewrites 41.2%

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

   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. 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 35.0%

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

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   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 36.1%

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

      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 39.2%

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

   6. Applied rewrites 39.2%

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

Alternative 11: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{t\_1}{\sqrt{t\_3 + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.71:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.7:\\ \;\;\;\;\frac{\sin th}{\sqrt{t\_3}} \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (* (sin th) (copysign 1.0 t_1)))
       (t_3 (pow (sin kx) 2.0))
       (t_4 (/ t_1 (sqrt (+ t_3 (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_4 -0.71)
     t_2
     (if (<= t_4 0.7)
       (* (/ (sin th) (sqrt t_3)) t_1)
       (if (<= t_4 2.0)
         t_2
         (* (sin th) (/ (fabs ky) (fabs (sin kx))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(th) * copysign(1.0, t_1);
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = t_1 / sqrt((t_3 + pow(t_1, 2.0)));
	double tmp;
	if (t_4 <= -0.71) {
		tmp = t_2;
	} else if (t_4 <= 0.7) {
		tmp = (sin(th) / sqrt(t_3)) * t_1;
	} else if (t_4 <= 2.0) {
		tmp = t_2;
	} else {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	}
	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 = Math.sin(th) * Math.copySign(1.0, t_1);
	double t_3 = Math.pow(Math.sin(kx), 2.0);
	double t_4 = t_1 / Math.sqrt((t_3 + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_4 <= -0.71) {
		tmp = t_2;
	} else if (t_4 <= 0.7) {
		tmp = (Math.sin(th) / Math.sqrt(t_3)) * t_1;
	} else if (t_4 <= 2.0) {
		tmp = t_2;
	} else {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(Math.sin(kx)));
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.sin(th) * math.copysign(1.0, t_1)
	t_3 = math.pow(math.sin(kx), 2.0)
	t_4 = t_1 / math.sqrt((t_3 + math.pow(t_1, 2.0)))
	tmp = 0
	if t_4 <= -0.71:
		tmp = t_2
	elif t_4 <= 0.7:
		tmp = (math.sin(th) / math.sqrt(t_3)) * t_1
	elif t_4 <= 2.0:
		tmp = t_2
	else:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(math.sin(kx)))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(sin(th) * copysign(1.0, t_1))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(t_1 / sqrt(Float64(t_3 + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.71)
		tmp = t_2;
	elseif (t_4 <= 0.7)
		tmp = Float64(Float64(sin(th) / sqrt(t_3)) * t_1);
	elseif (t_4 <= 2.0)
		tmp = t_2;
	else
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = sin(th) * (sign(t_1) * abs(1.0));
	t_3 = sin(kx) ^ 2.0;
	t_4 = t_1 / sqrt((t_3 + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_4 <= -0.71)
		tmp = t_2;
	elseif (t_4 <= 0.7)
		tmp = (sin(th) / sqrt(t_3)) * t_1;
	elseif (t_4 <= 2.0)
		tmp = t_2;
	else
		tmp = sin(th) * (abs(ky) / abs(sin(kx)));
	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[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[Sqrt[N[(t$95$3 + 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$4, -0.71], t$95$2, If[LessEqual[t$95$4, 0.7], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 2.0], t$95$2, N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.70999999999999996 or 0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 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-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.70999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 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-*.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 ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 41.1%

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

   6. Applied rewrites 41.1%

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

   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. 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 35.0%

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

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   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 36.1%

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

      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 39.2%

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

   6. Applied rewrites 39.2%

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

Alternative 12: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ t_3 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_4 := \sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.04:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (* (sin th) (copysign 1.0 t_1)))
       (t_3 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
       (t_4 (* (sin th) (/ (fabs ky) (fabs (sin kx))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_3 -0.04)
     t_2
     (if (<= t_3 5e-8) t_4 (if (<= t_3 2.0) t_2 t_4))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(th) * copysign(1.0, t_1);
	double t_3 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_4 = sin(th) * (fabs(ky) / fabs(sin(kx)));
	double tmp;
	if (t_3 <= -0.04) {
		tmp = t_2;
	} else if (t_3 <= 5e-8) {
		tmp = t_4;
	} else if (t_3 <= 2.0) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	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 = Math.sin(th) * Math.copySign(1.0, t_1);
	double t_3 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_4 = Math.sin(th) * (Math.abs(ky) / Math.abs(Math.sin(kx)));
	double tmp;
	if (t_3 <= -0.04) {
		tmp = t_2;
	} else if (t_3 <= 5e-8) {
		tmp = t_4;
	} else if (t_3 <= 2.0) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.sin(th) * math.copysign(1.0, t_1)
	t_3 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_4 = math.sin(th) * (math.fabs(ky) / math.fabs(math.sin(kx)))
	tmp = 0
	if t_3 <= -0.04:
		tmp = t_2
	elif t_3 <= 5e-8:
		tmp = t_4
	elif t_3 <= 2.0:
		tmp = t_2
	else:
		tmp = t_4
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(sin(th) * copysign(1.0, t_1))
	t_3 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_4 = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))))
	tmp = 0.0
	if (t_3 <= -0.04)
		tmp = t_2;
	elseif (t_3 <= 5e-8)
		tmp = t_4;
	elseif (t_3 <= 2.0)
		tmp = t_2;
	else
		tmp = t_4;
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = sin(th) * (sign(t_1) * abs(1.0));
	t_3 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_4 = sin(th) * (abs(ky) / abs(sin(kx)));
	tmp = 0.0;
	if (t_3 <= -0.04)
		tmp = t_2;
	elseif (t_3 <= 5e-8)
		tmp = t_4;
	elseif (t_3 <= 2.0)
		tmp = t_2;
	else
		tmp = t_4;
	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[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $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[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $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$3, -0.04], t$95$2, If[LessEqual[t$95$3, 5e-8], t$95$4, If[LessEqual[t$95$3, 2.0], t$95$2, t$95$4]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.040000000000000001 or 4.9999999999999998e-8 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.040000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.9999999999999998e-8 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 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 35.0%

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

   4. Applied rewrites 35.0%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   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 36.1%

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

      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 39.2%

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

   6. Applied rewrites 39.2%

      \[\leadsto \sin th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ t_3 := \frac{t\_1}{\sqrt{{\sin \left(\left|kx\right|\right)}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.04:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left|ky\right| \cdot \sin th}{\left|kx\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (* (sin th) (copysign 1.0 t_1)))
       (t_3
        (/ t_1 (sqrt (+ (pow (sin (fabs kx)) 2.0) (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_3 -0.04)
     t_2
     (if (<= t_3 2e-10) (/ (* (fabs ky) (sin th)) (fabs kx)) t_2)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(th) * copysign(1.0, t_1);
	double t_3 = t_1 / sqrt((pow(sin(fabs(kx)), 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= -0.04) {
		tmp = t_2;
	} else if (t_3 <= 2e-10) {
		tmp = (fabs(ky) * sin(th)) / fabs(kx);
	} else {
		tmp = t_2;
	}
	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 = Math.sin(th) * Math.copySign(1.0, t_1);
	double t_3 = t_1 / Math.sqrt((Math.pow(Math.sin(Math.abs(kx)), 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= -0.04) {
		tmp = t_2;
	} else if (t_3 <= 2e-10) {
		tmp = (Math.abs(ky) * Math.sin(th)) / Math.abs(kx);
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.sin(th) * math.copysign(1.0, t_1)
	t_3 = t_1 / math.sqrt((math.pow(math.sin(math.fabs(kx)), 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_3 <= -0.04:
		tmp = t_2
	elif t_3 <= 2e-10:
		tmp = (math.fabs(ky) * math.sin(th)) / math.fabs(kx)
	else:
		tmp = t_2
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(sin(th) * copysign(1.0, t_1))
	t_3 = Float64(t_1 / sqrt(Float64((sin(abs(kx)) ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.04)
		tmp = t_2;
	elseif (t_3 <= 2e-10)
		tmp = Float64(Float64(abs(ky) * sin(th)) / abs(kx));
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = sin(th) * (sign(t_1) * abs(1.0));
	t_3 = t_1 / sqrt(((sin(abs(kx)) ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -0.04)
		tmp = t_2;
	elseif (t_3 <= 2e-10)
		tmp = (abs(ky) * sin(th)) / abs(kx);
	else
		tmp = t_2;
	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[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.04], t$95$2, If[LessEqual[t$95$3, 2e-10], N[(N[(N[Abs[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision], t$95$2]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.040000000000000001 or 2.0000000000000001e-10 < (/.f64 (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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 41.3%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-fabs.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin ky \cdot \color{blue}{\sin th}\right) \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

      13. lower-*.f64 49.9%

          \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   6. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left|\sin ky\right|} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\left|\sin ky\right|}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\left|\sin ky\right|} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\color{blue}{\left|\sin ky\right|}} \]

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-fabs.f64 N/A

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

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

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-sin.f64 N/A

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

      19. pow2 N/A

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

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

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

      21. lift-fabs.f64 N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if -0.040000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.0000000000000001e-10

   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 35.0%

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

   4. Applied rewrites 35.0%

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

   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 15.9%

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

   7. Applied rewrites 15.9%

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 43.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left|ky\right| \cdot \sin th}{\sqrt{{kx}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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-10)
     (/ (* (fabs ky) (sin th)) (sqrt (pow kx 2.0)))
     (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)))) <= 2e-10) {
		tmp = (fabs(ky) * sin(th)) / sqrt(pow(kx, 2.0));
	} 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)))) <= 2e-10) {
		tmp = (Math.abs(ky) * Math.sin(th)) / Math.sqrt(Math.pow(kx, 2.0));
	} 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)))) <= 2e-10:
		tmp = (math.fabs(ky) * math.sin(th)) / math.sqrt(math.pow(kx, 2.0))
	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)))) <= 2e-10)
		tmp = Float64(Float64(abs(ky) * sin(th)) / sqrt((kx ^ 2.0)));
	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)))) <= 2e-10)
		tmp = (abs(ky) * sin(th)) / sqrt((kx ^ 2.0));
	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], 2e-10], N[(N[(N[Abs[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Power[kx, 2.0], $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))))) < 2.0000000000000001e-10

   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 35.0%

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

   4. Applied rewrites 35.0%

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

   5. Taylor expanded in kx around 0

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

      1. lower-pow.f64 18.0%

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

   7. Applied rewrites 18.0%

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

   if 2.0000000000000001e-10 < (/.f64 (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 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 43.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin \left(\left|kx\right|\right)}^{2} + {t\_1}^{2}}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left|ky\right| \cdot \sin th}{\left|kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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-10)
     (/ (* (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-10) {
		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-10) {
		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-10:
		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-10)
		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-10)
		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-10], 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))))) < 2.0000000000000001e-10

   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 35.0%

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

   4. Applied rewrites 35.0%

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

   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 15.9%

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

   7. Applied rewrites 15.9%

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

   if 2.0000000000000001e-10 < (/.f64 (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 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 36.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 3.7 \cdot 10^{-22}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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))))
        3.7e-22)
     (* -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)))) <= 3.7e-22) {
		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)))) <= 3.7e-22) {
		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)))) <= 3.7e-22:
		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)))) <= 3.7e-22)
		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)))) <= 3.7e-22)
		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], 3.7e-22], 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))))) < 3.7e-22

   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 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

   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 13.3%

         \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

   10. Applied rewrites 13.3%

       \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]

   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 10.9%

          \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   13. Applied rewrites 10.9%

       \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   if 3.7e-22 < (/.f64 (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 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 22.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 8 \cdot 10^{-22}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th \cdot th, -0.16666666666666666 \cdot th, th\right)\\ \end{array} \end{array} \]

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)))) 8e-22)
     (* -0.16666666666666666 (pow th 3.0))
     (fma (* th th) (* -0.16666666666666666 th) 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)))) <= 8e-22) {
		tmp = -0.16666666666666666 * pow(th, 3.0);
	} else {
		tmp = fma((th * th), (-0.16666666666666666 * th), th);
	}
	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)))) <= 8e-22)
		tmp = Float64(-0.16666666666666666 * (th ^ 3.0));
	else
		tmp = fma(Float64(th * th), Float64(-0.16666666666666666 * th), th);
	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], 8e-22], N[(-0.16666666666666666 * N[Power[th, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(th * th), $MachinePrecision] * N[(-0.16666666666666666 * th), $MachinePrecision] + 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))))) < 8.0000000000000004e-22

   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 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

   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 13.3%

         \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

   10. Applied rewrites 13.3%

       \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]

   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 10.9%

          \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   13. Applied rewrites 10.9%

       \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   if 8.0000000000000004e-22 < (/.f64 (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 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

   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 13.3%

         \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

   10. Applied rewrites 13.3%

       \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

          \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th + 1 \cdot th \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th + 1 \cdot th \]

       6. *-commutative N/A

          \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th + 1 \cdot th \]

       7. associate-*l* N/A

          \[\leadsto {th}^{2} \cdot \left(\frac{-1}{6} \cdot th\right) + 1 \cdot th \]

       8. *-lft-identity N/A

          \[\leadsto {th}^{2} \cdot \left(\frac{-1}{6} \cdot th\right) + th \]

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 13.3%

           \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666 \cdot th, th\right) \]

   12. Applied rewrites 13.3%

       \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666 \cdot th, th\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 15.6% accurate, 10.3× speedup

Math

\[\mathsf{copysign}\left(1, ky\right) \cdot \mathsf{fma}\left(th \cdot th, -0.16666666666666666 \cdot th, th\right) \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (* (copysign 1.0 ky) (fma (* th th) (* -0.16666666666666666 th) th)))

C

double code(double kx, double ky, double th) {
	return copysign(1.0, ky) * fma((th * th), (-0.16666666666666666 * th), th);
}

Julia

function code(kx, ky, th)
	return Float64(copysign(1.0, ky) * fma(Float64(th * th), Float64(-0.16666666666666666 * th), th))
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[(th * th), $MachinePrecision] * N[(-0.16666666666666666 * th), $MachinePrecision] + th), $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 23.7%

      \[\leadsto \sin th \]

7. Applied rewrites 23.7%

   \[\leadsto \sin th \]

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 13.3%

      \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

10. Applied rewrites 13.3%

    \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. +-commutative N/A

       \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \]

    4. distribute-rgt-in N/A

       \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th + 1 \cdot th \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th + 1 \cdot th \]

    6. *-commutative N/A

       \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th + 1 \cdot th \]

    7. associate-*l* N/A

       \[\leadsto {th}^{2} \cdot \left(\frac{-1}{6} \cdot th\right) + 1 \cdot th \]

    8. *-lft-identity N/A

       \[\leadsto {th}^{2} \cdot \left(\frac{-1}{6} \cdot th\right) + th \]

    9. lower-fma.f64 N/A

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

    10. lift-pow.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-*.f64 13.3%

        \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666 \cdot th, th\right) \]

12. Applied rewrites 13.3%

    \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666 \cdot th, th\right) \]
13. Add Preprocessing

Alternative 19: 15.6% 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 23.7%

      \[\leadsto \sin th \]

7. Applied rewrites 23.7%

   \[\leadsto \sin th \]

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 13.3%

      \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

10. Applied rewrites 13.3%

    \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]
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 13.3%

       \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \]

12. Applied rewrites 13.3%

    \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025213 
(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)))