Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 8.4s

Alternatives: 24

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: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \frac{t\_2}{\mathsf{hypot}\left(t\_2, \sin kx\right)} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\\ t_4 := {t\_2}^{2}\\ t_5 := {\sin kx}^{2}\\ t_6 := \frac{t\_2}{\sqrt{t\_5 + t\_4}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_6 \leq -0.9998:\\ \;\;\;\;\frac{\sin th \cdot t\_2}{\mathsf{hypot}\left(kx \cdot \left(1 + -0.16666666666666666 \cdot {kx}^{2}\right), t\_2\right)}\\ \mathbf{elif}\;t\_6 \leq -0.3:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_5}} \cdot \sin th\\ \mathbf{elif}\;t\_6 \leq 0.9990377408563715:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_6 \leq 1:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_4}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1
        (*
         (fabs ky)
         (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0)))))
       (t_2 (sin (fabs ky)))
       (t_3
        (*
         (/ t_2 (hypot t_2 (sin kx)))
         (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0))))))
       (t_4 (pow t_2 2.0))
       (t_5 (pow (sin kx) 2.0))
       (t_6 (/ t_2 (sqrt (+ t_5 t_4)))))
  (*
   (copysign 1.0 ky)
   (if (<= t_6 -0.9998)
     (/
      (* (sin th) t_2)
      (hypot
       (* kx (+ 1.0 (* -0.16666666666666666 (pow kx 2.0))))
       t_2))
     (if (<= t_6 -0.3)
       t_3
       (if (<= t_6 5e-15)
         (* (/ t_2 (sqrt t_5)) (sin th))
         (if (<= t_6 0.9990377408563715)
           t_3
           (if (<= t_6 1.0)
             (* (/ t_2 (sqrt t_4)) (sin th))
             (* (/ t_1 (hypot t_1 (sin kx))) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double t_2 = sin(fabs(ky));
	double t_3 = (t_2 / hypot(t_2, sin(kx))) * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0))));
	double t_4 = pow(t_2, 2.0);
	double t_5 = pow(sin(kx), 2.0);
	double t_6 = t_2 / sqrt((t_5 + t_4));
	double tmp;
	if (t_6 <= -0.9998) {
		tmp = (sin(th) * t_2) / hypot((kx * (1.0 + (-0.16666666666666666 * pow(kx, 2.0)))), t_2);
	} else if (t_6 <= -0.3) {
		tmp = t_3;
	} else if (t_6 <= 5e-15) {
		tmp = (t_2 / sqrt(t_5)) * sin(th);
	} else if (t_6 <= 0.9990377408563715) {
		tmp = t_3;
	} else if (t_6 <= 1.0) {
		tmp = (t_2 / sqrt(t_4)) * sin(th);
	} else {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	}
	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 = (t_2 / Math.hypot(t_2, Math.sin(kx))) * (th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0))));
	double t_4 = Math.pow(t_2, 2.0);
	double t_5 = Math.pow(Math.sin(kx), 2.0);
	double t_6 = t_2 / Math.sqrt((t_5 + t_4));
	double tmp;
	if (t_6 <= -0.9998) {
		tmp = (Math.sin(th) * t_2) / Math.hypot((kx * (1.0 + (-0.16666666666666666 * Math.pow(kx, 2.0)))), t_2);
	} else if (t_6 <= -0.3) {
		tmp = t_3;
	} else if (t_6 <= 5e-15) {
		tmp = (t_2 / Math.sqrt(t_5)) * Math.sin(th);
	} else if (t_6 <= 0.9990377408563715) {
		tmp = t_3;
	} else if (t_6 <= 1.0) {
		tmp = (t_2 / Math.sqrt(t_4)) * Math.sin(th);
	} else {
		tmp = (t_1 / Math.hypot(t_1, Math.sin(kx))) * Math.sin(th);
	}
	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 = (t_2 / math.hypot(t_2, math.sin(kx))) * (th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0))))
	t_4 = math.pow(t_2, 2.0)
	t_5 = math.pow(math.sin(kx), 2.0)
	t_6 = t_2 / math.sqrt((t_5 + t_4))
	tmp = 0
	if t_6 <= -0.9998:
		tmp = (math.sin(th) * t_2) / math.hypot((kx * (1.0 + (-0.16666666666666666 * math.pow(kx, 2.0)))), t_2)
	elif t_6 <= -0.3:
		tmp = t_3
	elif t_6 <= 5e-15:
		tmp = (t_2 / math.sqrt(t_5)) * math.sin(th)
	elif t_6 <= 0.9990377408563715:
		tmp = t_3
	elif t_6 <= 1.0:
		tmp = (t_2 / math.sqrt(t_4)) * math.sin(th)
	else:
		tmp = (t_1 / math.hypot(t_1, math.sin(kx))) * math.sin(th)
	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 = Float64(Float64(t_2 / hypot(t_2, sin(kx))) * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))))
	t_4 = t_2 ^ 2.0
	t_5 = sin(kx) ^ 2.0
	t_6 = Float64(t_2 / sqrt(Float64(t_5 + t_4)))
	tmp = 0.0
	if (t_6 <= -0.9998)
		tmp = Float64(Float64(sin(th) * t_2) / hypot(Float64(kx * Float64(1.0 + Float64(-0.16666666666666666 * (kx ^ 2.0)))), t_2));
	elseif (t_6 <= -0.3)
		tmp = t_3;
	elseif (t_6 <= 5e-15)
		tmp = Float64(Float64(t_2 / sqrt(t_5)) * sin(th));
	elseif (t_6 <= 0.9990377408563715)
		tmp = t_3;
	elseif (t_6 <= 1.0)
		tmp = Float64(Float64(t_2 / sqrt(t_4)) * sin(th));
	else
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	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 / hypot(t_2, sin(kx))) * (th * (1.0 + (-0.16666666666666666 * (th ^ 2.0))));
	t_4 = t_2 ^ 2.0;
	t_5 = sin(kx) ^ 2.0;
	t_6 = t_2 / sqrt((t_5 + t_4));
	tmp = 0.0;
	if (t_6 <= -0.9998)
		tmp = (sin(th) * t_2) / hypot((kx * (1.0 + (-0.16666666666666666 * (kx ^ 2.0)))), t_2);
	elseif (t_6 <= -0.3)
		tmp = t_3;
	elseif (t_6 <= 5e-15)
		tmp = (t_2 / sqrt(t_5)) * sin(th);
	elseif (t_6 <= 0.9990377408563715)
		tmp = t_3;
	elseif (t_6 <= 1.0)
		tmp = (t_2 / sqrt(t_4)) * sin(th);
	else
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	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[(N[(t$95$2 / N[Sqrt[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 / N[Sqrt[N[(t$95$5 + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$6, -0.9998], N[(N[(N[Sin[th], $MachinePrecision] * t$95$2), $MachinePrecision] / N[Sqrt[N[(kx * N[(1.0 + N[(-0.16666666666666666 * N[Power[kx, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, -0.3], t$95$3, If[LessEqual[t$95$6, 5e-15], N[(N[(t$95$2 / N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 0.9990377408563715], t$95$3, If[LessEqual[t$95$6, 1.0], N[(N[(t$95$2 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 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))))) < -0.99980000000000002

   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. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 96.1%

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.4%

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

   6. Applied rewrites 54.4%

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

   if -0.99980000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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 50.8%

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

   6. Applied rewrites 50.8%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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 41.7%

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

   4. Applied rewrites 41.7%

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

   if 0.99903774085637154 < (/.f64 (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

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6%

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

   4. Applied rewrites 40.6%

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

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.3%

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.8%

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

   9. Applied rewrites 54.8%

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

Alternative 3: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \left(-t\_1\right) \cdot \left(\frac{-1}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\right)\\ t_3 := \left(\frac{1}{\mathsf{hypot}\left(kx \cdot \left(1 + -0.16666666666666666 \cdot {kx}^{2}\right), t\_1\right)} \cdot t\_1\right) \cdot \sin th\\ t_4 := {\sin kx}^{2}\\ t_5 := \frac{t\_1}{\sqrt{t\_4 + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -0.9998:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq -0.3:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_4}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.9990377408563715:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2
        (*
         (- t_1)
         (*
          (/ -1.0 (hypot (sin kx) t_1))
          (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0)))))))
       (t_3
        (*
         (*
          (/
           1.0
           (hypot
            (* kx (+ 1.0 (* -0.16666666666666666 (pow kx 2.0))))
            t_1))
          t_1)
         (sin th)))
       (t_4 (pow (sin kx) 2.0))
       (t_5 (/ t_1 (sqrt (+ t_4 (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -0.9998)
     t_3
     (if (<= t_5 -0.3)
       t_2
       (if (<= t_5 5e-15)
         (* (/ t_1 (sqrt t_4)) (sin th))
         (if (<= t_5 0.9990377408563715) t_2 t_3)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = -t_1 * ((-1.0 / hypot(sin(kx), t_1)) * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0)))));
	double t_3 = ((1.0 / hypot((kx * (1.0 + (-0.16666666666666666 * pow(kx, 2.0)))), t_1)) * t_1) * sin(th);
	double t_4 = pow(sin(kx), 2.0);
	double t_5 = t_1 / sqrt((t_4 + pow(t_1, 2.0)));
	double tmp;
	if (t_5 <= -0.9998) {
		tmp = t_3;
	} else if (t_5 <= -0.3) {
		tmp = t_2;
	} else if (t_5 <= 5e-15) {
		tmp = (t_1 / sqrt(t_4)) * sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = -t_1 * ((-1.0 / Math.hypot(Math.sin(kx), t_1)) * (th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0)))));
	double t_3 = ((1.0 / Math.hypot((kx * (1.0 + (-0.16666666666666666 * Math.pow(kx, 2.0)))), t_1)) * t_1) * Math.sin(th);
	double t_4 = Math.pow(Math.sin(kx), 2.0);
	double t_5 = t_1 / Math.sqrt((t_4 + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_5 <= -0.9998) {
		tmp = t_3;
	} else if (t_5 <= -0.3) {
		tmp = t_2;
	} else if (t_5 <= 5e-15) {
		tmp = (t_1 / Math.sqrt(t_4)) * Math.sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = -t_1 * ((-1.0 / math.hypot(math.sin(kx), t_1)) * (th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0)))))
	t_3 = ((1.0 / math.hypot((kx * (1.0 + (-0.16666666666666666 * math.pow(kx, 2.0)))), t_1)) * t_1) * math.sin(th)
	t_4 = math.pow(math.sin(kx), 2.0)
	t_5 = t_1 / math.sqrt((t_4 + math.pow(t_1, 2.0)))
	tmp = 0
	if t_5 <= -0.9998:
		tmp = t_3
	elif t_5 <= -0.3:
		tmp = t_2
	elif t_5 <= 5e-15:
		tmp = (t_1 / math.sqrt(t_4)) * math.sin(th)
	elif t_5 <= 0.9990377408563715:
		tmp = t_2
	else:
		tmp = t_3
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(-t_1) * Float64(Float64(-1.0 / hypot(sin(kx), t_1)) * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0))))))
	t_3 = Float64(Float64(Float64(1.0 / hypot(Float64(kx * Float64(1.0 + Float64(-0.16666666666666666 * (kx ^ 2.0)))), t_1)) * t_1) * sin(th))
	t_4 = sin(kx) ^ 2.0
	t_5 = Float64(t_1 / sqrt(Float64(t_4 + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_5 <= -0.9998)
		tmp = t_3;
	elseif (t_5 <= -0.3)
		tmp = t_2;
	elseif (t_5 <= 5e-15)
		tmp = Float64(Float64(t_1 / sqrt(t_4)) * sin(th));
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = -t_1 * ((-1.0 / hypot(sin(kx), t_1)) * (th * (1.0 + (-0.16666666666666666 * (th ^ 2.0)))));
	t_3 = ((1.0 / hypot((kx * (1.0 + (-0.16666666666666666 * (kx ^ 2.0)))), t_1)) * t_1) * sin(th);
	t_4 = sin(kx) ^ 2.0;
	t_5 = t_1 / sqrt((t_4 + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_5 <= -0.9998)
		tmp = t_3;
	elseif (t_5 <= -0.3)
		tmp = t_2;
	elseif (t_5 <= 5e-15)
		tmp = (t_1 / sqrt(t_4)) * sin(th);
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[((-t$95$1) * N[(N[(-1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1.0 / 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] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 / N[Sqrt[N[(t$95$4 + 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$5, -0.9998], t$95$3, If[LessEqual[t$95$5, -0.3], t$95$2, If[LessEqual[t$95$5, 5e-15], N[(N[(t$95$1 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.9990377408563715], t$95$2, t$95$3]]]]), $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))))) < -0.99980000000000002 or 0.99903774085637154 < (/.f64 (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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 93.6%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 57.8%

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

   6. Applied rewrites 57.8%

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

   if -0.99980000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.7%

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

   6. Applied rewrites 50.7%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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 41.7%

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

   4. Applied rewrites 41.7%

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

Alternative 4: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\\ t_3 := \left(\frac{1}{\mathsf{hypot}\left(kx \cdot \left(1 + -0.16666666666666666 \cdot {kx}^{2}\right), t\_1\right)} \cdot t\_1\right) \cdot \sin th\\ t_4 := {\sin kx}^{2}\\ t_5 := \frac{t\_1}{\sqrt{t\_4 + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -0.9998:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq -0.3:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_4}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.9990377408563715:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2
        (*
         (/ t_1 (hypot t_1 (sin kx)))
         (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0))))))
       (t_3
        (*
         (*
          (/
           1.0
           (hypot
            (* kx (+ 1.0 (* -0.16666666666666666 (pow kx 2.0))))
            t_1))
          t_1)
         (sin th)))
       (t_4 (pow (sin kx) 2.0))
       (t_5 (/ t_1 (sqrt (+ t_4 (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -0.9998)
     t_3
     (if (<= t_5 -0.3)
       t_2
       (if (<= t_5 5e-15)
         (* (/ t_1 (sqrt t_4)) (sin th))
         (if (<= t_5 0.9990377408563715) t_2 t_3)))))))

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))) * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0))));
	double t_3 = ((1.0 / hypot((kx * (1.0 + (-0.16666666666666666 * pow(kx, 2.0)))), t_1)) * t_1) * sin(th);
	double t_4 = pow(sin(kx), 2.0);
	double t_5 = t_1 / sqrt((t_4 + pow(t_1, 2.0)));
	double tmp;
	if (t_5 <= -0.9998) {
		tmp = t_3;
	} else if (t_5 <= -0.3) {
		tmp = t_2;
	} else if (t_5 <= 5e-15) {
		tmp = (t_1 / sqrt(t_4)) * sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = (t_1 / Math.hypot(t_1, Math.sin(kx))) * (th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0))));
	double t_3 = ((1.0 / Math.hypot((kx * (1.0 + (-0.16666666666666666 * Math.pow(kx, 2.0)))), t_1)) * t_1) * Math.sin(th);
	double t_4 = Math.pow(Math.sin(kx), 2.0);
	double t_5 = t_1 / Math.sqrt((t_4 + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_5 <= -0.9998) {
		tmp = t_3;
	} else if (t_5 <= -0.3) {
		tmp = t_2;
	} else if (t_5 <= 5e-15) {
		tmp = (t_1 / Math.sqrt(t_4)) * Math.sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = (t_1 / math.hypot(t_1, math.sin(kx))) * (th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0))))
	t_3 = ((1.0 / math.hypot((kx * (1.0 + (-0.16666666666666666 * math.pow(kx, 2.0)))), t_1)) * t_1) * math.sin(th)
	t_4 = math.pow(math.sin(kx), 2.0)
	t_5 = t_1 / math.sqrt((t_4 + math.pow(t_1, 2.0)))
	tmp = 0
	if t_5 <= -0.9998:
		tmp = t_3
	elif t_5 <= -0.3:
		tmp = t_2
	elif t_5 <= 5e-15:
		tmp = (t_1 / math.sqrt(t_4)) * math.sin(th)
	elif t_5 <= 0.9990377408563715:
		tmp = t_2
	else:
		tmp = t_3
	return math.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))) * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))))
	t_3 = Float64(Float64(Float64(1.0 / hypot(Float64(kx * Float64(1.0 + Float64(-0.16666666666666666 * (kx ^ 2.0)))), t_1)) * t_1) * sin(th))
	t_4 = sin(kx) ^ 2.0
	t_5 = Float64(t_1 / sqrt(Float64(t_4 + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_5 <= -0.9998)
		tmp = t_3;
	elseif (t_5 <= -0.3)
		tmp = t_2;
	elseif (t_5 <= 5e-15)
		tmp = Float64(Float64(t_1 / sqrt(t_4)) * sin(th));
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = (t_1 / hypot(t_1, sin(kx))) * (th * (1.0 + (-0.16666666666666666 * (th ^ 2.0))));
	t_3 = ((1.0 / hypot((kx * (1.0 + (-0.16666666666666666 * (kx ^ 2.0)))), t_1)) * t_1) * sin(th);
	t_4 = sin(kx) ^ 2.0;
	t_5 = t_1 / sqrt((t_4 + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_5 <= -0.9998)
		tmp = t_3;
	elseif (t_5 <= -0.3)
		tmp = t_2;
	elseif (t_5 <= 5e-15)
		tmp = (t_1 / sqrt(t_4)) * sin(th);
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_2;
	else
		tmp = t_3;
	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[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1.0 / 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] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 / N[Sqrt[N[(t$95$4 + 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$5, -0.9998], t$95$3, If[LessEqual[t$95$5, -0.3], t$95$2, If[LessEqual[t$95$5, 5e-15], N[(N[(t$95$1 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.9990377408563715], t$95$2, t$95$3]]]]), $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))))) < -0.99980000000000002 or 0.99903774085637154 < (/.f64 (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. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 93.6%

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

      6. lift-sqrt.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 57.8%

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

   6. Applied rewrites 57.8%

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

   if -0.99980000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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 50.8%

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

   6. Applied rewrites 50.8%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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 41.7%

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

   4. Applied rewrites 41.7%

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

Alternative 5: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := {t\_2}^{2}\\ t_4 := {\sin kx}^{2}\\ t_5 := \frac{t\_2}{\sqrt{t\_4 + t\_3}}\\ t_6 := \frac{t\_2}{\mathsf{hypot}\left(t\_2, \sin kx\right)} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -0.9998:\\ \;\;\;\;\frac{t\_2}{\sqrt{{kx}^{2} + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq -0.3:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_4}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.9990377408563715:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq 1:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1
        (*
         (fabs ky)
         (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0)))))
       (t_2 (sin (fabs ky)))
       (t_3 (pow t_2 2.0))
       (t_4 (pow (sin kx) 2.0))
       (t_5 (/ t_2 (sqrt (+ t_4 t_3))))
       (t_6
        (*
         (/ t_2 (hypot t_2 (sin kx)))
         (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0)))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -0.9998)
     (* (/ t_2 (sqrt (+ (pow kx 2.0) t_3))) (sin th))
     (if (<= t_5 -0.3)
       t_6
       (if (<= t_5 5e-15)
         (* (/ t_2 (sqrt t_4)) (sin th))
         (if (<= t_5 0.9990377408563715)
           t_6
           (if (<= t_5 1.0)
             (* (/ t_2 (sqrt t_3)) (sin th))
             (* (/ t_1 (hypot t_1 (sin kx))) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double t_2 = sin(fabs(ky));
	double t_3 = pow(t_2, 2.0);
	double t_4 = pow(sin(kx), 2.0);
	double t_5 = t_2 / sqrt((t_4 + t_3));
	double t_6 = (t_2 / hypot(t_2, sin(kx))) * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0))));
	double tmp;
	if (t_5 <= -0.9998) {
		tmp = (t_2 / sqrt((pow(kx, 2.0) + t_3))) * sin(th);
	} else if (t_5 <= -0.3) {
		tmp = t_6;
	} else if (t_5 <= 5e-15) {
		tmp = (t_2 / sqrt(t_4)) * sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_6;
	} else if (t_5 <= 1.0) {
		tmp = (t_2 / sqrt(t_3)) * sin(th);
	} else {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	}
	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 = t_2 / Math.sqrt((t_4 + t_3));
	double t_6 = (t_2 / Math.hypot(t_2, Math.sin(kx))) * (th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0))));
	double tmp;
	if (t_5 <= -0.9998) {
		tmp = (t_2 / Math.sqrt((Math.pow(kx, 2.0) + t_3))) * Math.sin(th);
	} else if (t_5 <= -0.3) {
		tmp = t_6;
	} else if (t_5 <= 5e-15) {
		tmp = (t_2 / Math.sqrt(t_4)) * Math.sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_6;
	} else if (t_5 <= 1.0) {
		tmp = (t_2 / Math.sqrt(t_3)) * Math.sin(th);
	} else {
		tmp = (t_1 / Math.hypot(t_1, Math.sin(kx))) * Math.sin(th);
	}
	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 = t_2 / math.sqrt((t_4 + t_3))
	t_6 = (t_2 / math.hypot(t_2, math.sin(kx))) * (th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0))))
	tmp = 0
	if t_5 <= -0.9998:
		tmp = (t_2 / math.sqrt((math.pow(kx, 2.0) + t_3))) * math.sin(th)
	elif t_5 <= -0.3:
		tmp = t_6
	elif t_5 <= 5e-15:
		tmp = (t_2 / math.sqrt(t_4)) * math.sin(th)
	elif t_5 <= 0.9990377408563715:
		tmp = t_6
	elif t_5 <= 1.0:
		tmp = (t_2 / math.sqrt(t_3)) * math.sin(th)
	else:
		tmp = (t_1 / math.hypot(t_1, math.sin(kx))) * math.sin(th)
	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 = Float64(t_2 / sqrt(Float64(t_4 + t_3)))
	t_6 = Float64(Float64(t_2 / hypot(t_2, sin(kx))) * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))))
	tmp = 0.0
	if (t_5 <= -0.9998)
		tmp = Float64(Float64(t_2 / sqrt(Float64((kx ^ 2.0) + t_3))) * sin(th));
	elseif (t_5 <= -0.3)
		tmp = t_6;
	elseif (t_5 <= 5e-15)
		tmp = Float64(Float64(t_2 / sqrt(t_4)) * sin(th));
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_6;
	elseif (t_5 <= 1.0)
		tmp = Float64(Float64(t_2 / sqrt(t_3)) * sin(th));
	else
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

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 = t_2 / sqrt((t_4 + t_3));
	t_6 = (t_2 / hypot(t_2, sin(kx))) * (th * (1.0 + (-0.16666666666666666 * (th ^ 2.0))));
	tmp = 0.0;
	if (t_5 <= -0.9998)
		tmp = (t_2 / sqrt(((kx ^ 2.0) + t_3))) * sin(th);
	elseif (t_5 <= -0.3)
		tmp = t_6;
	elseif (t_5 <= 5e-15)
		tmp = (t_2 / sqrt(t_4)) * sin(th);
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_6;
	elseif (t_5 <= 1.0)
		tmp = (t_2 / sqrt(t_3)) * sin(th);
	else
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	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[(t$95$2 / N[Sqrt[N[(t$95$4 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$2 / N[Sqrt[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $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, -0.9998], 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], If[LessEqual[t$95$5, -0.3], t$95$6, If[LessEqual[t$95$5, 5e-15], N[(N[(t$95$2 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.9990377408563715], t$95$6, If[LessEqual[t$95$5, 1.0], N[(N[(t$95$2 / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 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))))) < -0.99980000000000002

   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 52.2%

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

   4. Applied rewrites 52.2%

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

   if -0.99980000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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 50.8%

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

   6. Applied rewrites 50.8%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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 41.7%

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

   4. Applied rewrites 41.7%

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

   if 0.99903774085637154 < (/.f64 (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

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6%

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

   4. Applied rewrites 40.6%

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

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.3%

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.8%

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

   9. Applied rewrites 54.8%

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

Alternative 6: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := {t\_2}^{2}\\ t_4 := {\sin kx}^{2}\\ t_5 := \frac{t\_2}{\sqrt{t\_4 + t\_3}}\\ t_6 := \frac{th \cdot t\_2}{\mathsf{hypot}\left(\sin kx, t\_2\right)}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -0.9998:\\ \;\;\;\;\frac{t\_2}{\sqrt{{kx}^{2} + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq -0.3:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_4}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.9990377408563715:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq 1:\\ \;\;\;\;\frac{t\_2}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1
        (*
         (fabs ky)
         (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0)))))
       (t_2 (sin (fabs ky)))
       (t_3 (pow t_2 2.0))
       (t_4 (pow (sin kx) 2.0))
       (t_5 (/ t_2 (sqrt (+ t_4 t_3))))
       (t_6 (/ (* th t_2) (hypot (sin kx) t_2))))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -0.9998)
     (* (/ t_2 (sqrt (+ (pow kx 2.0) t_3))) (sin th))
     (if (<= t_5 -0.3)
       t_6
       (if (<= t_5 5e-15)
         (* (/ t_2 (sqrt t_4)) (sin th))
         (if (<= t_5 0.9990377408563715)
           t_6
           (if (<= t_5 1.0)
             (* (/ t_2 (sqrt t_3)) (sin th))
             (* (/ t_1 (hypot t_1 (sin kx))) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double t_2 = sin(fabs(ky));
	double t_3 = pow(t_2, 2.0);
	double t_4 = pow(sin(kx), 2.0);
	double t_5 = t_2 / sqrt((t_4 + t_3));
	double t_6 = (th * t_2) / hypot(sin(kx), t_2);
	double tmp;
	if (t_5 <= -0.9998) {
		tmp = (t_2 / sqrt((pow(kx, 2.0) + t_3))) * sin(th);
	} else if (t_5 <= -0.3) {
		tmp = t_6;
	} else if (t_5 <= 5e-15) {
		tmp = (t_2 / sqrt(t_4)) * sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_6;
	} else if (t_5 <= 1.0) {
		tmp = (t_2 / sqrt(t_3)) * sin(th);
	} else {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	}
	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 = t_2 / Math.sqrt((t_4 + t_3));
	double t_6 = (th * t_2) / Math.hypot(Math.sin(kx), t_2);
	double tmp;
	if (t_5 <= -0.9998) {
		tmp = (t_2 / Math.sqrt((Math.pow(kx, 2.0) + t_3))) * Math.sin(th);
	} else if (t_5 <= -0.3) {
		tmp = t_6;
	} else if (t_5 <= 5e-15) {
		tmp = (t_2 / Math.sqrt(t_4)) * Math.sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_6;
	} else if (t_5 <= 1.0) {
		tmp = (t_2 / Math.sqrt(t_3)) * Math.sin(th);
	} else {
		tmp = (t_1 / Math.hypot(t_1, Math.sin(kx))) * Math.sin(th);
	}
	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 = t_2 / math.sqrt((t_4 + t_3))
	t_6 = (th * t_2) / math.hypot(math.sin(kx), t_2)
	tmp = 0
	if t_5 <= -0.9998:
		tmp = (t_2 / math.sqrt((math.pow(kx, 2.0) + t_3))) * math.sin(th)
	elif t_5 <= -0.3:
		tmp = t_6
	elif t_5 <= 5e-15:
		tmp = (t_2 / math.sqrt(t_4)) * math.sin(th)
	elif t_5 <= 0.9990377408563715:
		tmp = t_6
	elif t_5 <= 1.0:
		tmp = (t_2 / math.sqrt(t_3)) * math.sin(th)
	else:
		tmp = (t_1 / math.hypot(t_1, math.sin(kx))) * math.sin(th)
	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 = Float64(t_2 / sqrt(Float64(t_4 + t_3)))
	t_6 = Float64(Float64(th * t_2) / hypot(sin(kx), t_2))
	tmp = 0.0
	if (t_5 <= -0.9998)
		tmp = Float64(Float64(t_2 / sqrt(Float64((kx ^ 2.0) + t_3))) * sin(th));
	elseif (t_5 <= -0.3)
		tmp = t_6;
	elseif (t_5 <= 5e-15)
		tmp = Float64(Float64(t_2 / sqrt(t_4)) * sin(th));
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_6;
	elseif (t_5 <= 1.0)
		tmp = Float64(Float64(t_2 / sqrt(t_3)) * sin(th));
	else
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

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 = t_2 / sqrt((t_4 + t_3));
	t_6 = (th * t_2) / hypot(sin(kx), t_2);
	tmp = 0.0;
	if (t_5 <= -0.9998)
		tmp = (t_2 / sqrt(((kx ^ 2.0) + t_3))) * sin(th);
	elseif (t_5 <= -0.3)
		tmp = t_6;
	elseif (t_5 <= 5e-15)
		tmp = (t_2 / sqrt(t_4)) * sin(th);
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_6;
	elseif (t_5 <= 1.0)
		tmp = (t_2 / sqrt(t_3)) * sin(th);
	else
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	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[(t$95$2 / N[Sqrt[N[(t$95$4 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(th * t$95$2), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, -0.9998], 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], If[LessEqual[t$95$5, -0.3], t$95$6, If[LessEqual[t$95$5, 5e-15], N[(N[(t$95$2 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.9990377408563715], t$95$6, If[LessEqual[t$95$5, 1.0], N[(N[(t$95$2 / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 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))))) < -0.99980000000000002

   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 52.2%

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

   4. Applied rewrites 52.2%

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

   if -0.99980000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 96.1%

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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 41.7%

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

   4. Applied rewrites 41.7%

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

   if 0.99903774085637154 < (/.f64 (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

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6%

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

   4. Applied rewrites 40.6%

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

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.3%

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.8%

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

   9. Applied rewrites 54.8%

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

Alternative 7: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := {t\_1}^{2}\\ t_3 := \frac{t\_1}{\sqrt{t\_2}} \cdot \sin th\\ t_4 := {\sin kx}^{2}\\ t_5 := \frac{t\_1}{\sqrt{t\_4 + t\_2}}\\ t_6 := \frac{th \cdot t\_1}{\mathsf{hypot}\left(\sin kx, t\_1\right)}\\ t_7 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -0.9948:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq -0.3:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_4}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.9990377408563715:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq 1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_7}{\mathsf{hypot}\left(t\_7, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (pow t_1 2.0))
       (t_3 (* (/ t_1 (sqrt t_2)) (sin th)))
       (t_4 (pow (sin kx) 2.0))
       (t_5 (/ t_1 (sqrt (+ t_4 t_2))))
       (t_6 (/ (* th t_1) (hypot (sin kx) t_1)))
       (t_7
        (*
         (fabs ky)
         (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -0.9948)
     t_3
     (if (<= t_5 -0.3)
       t_6
       (if (<= t_5 5e-15)
         (* (/ t_1 (sqrt t_4)) (sin th))
         (if (<= t_5 0.9990377408563715)
           t_6
           (if (<= t_5 1.0)
             t_3
             (* (/ t_7 (hypot t_7 (sin kx))) (sin th))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = pow(t_1, 2.0);
	double t_3 = (t_1 / sqrt(t_2)) * sin(th);
	double t_4 = pow(sin(kx), 2.0);
	double t_5 = t_1 / sqrt((t_4 + t_2));
	double t_6 = (th * t_1) / hypot(sin(kx), t_1);
	double t_7 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double tmp;
	if (t_5 <= -0.9948) {
		tmp = t_3;
	} else if (t_5 <= -0.3) {
		tmp = t_6;
	} else if (t_5 <= 5e-15) {
		tmp = (t_1 / sqrt(t_4)) * sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_6;
	} else if (t_5 <= 1.0) {
		tmp = t_3;
	} else {
		tmp = (t_7 / hypot(t_7, sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = Math.pow(t_1, 2.0);
	double t_3 = (t_1 / Math.sqrt(t_2)) * Math.sin(th);
	double t_4 = Math.pow(Math.sin(kx), 2.0);
	double t_5 = t_1 / Math.sqrt((t_4 + t_2));
	double t_6 = (th * t_1) / Math.hypot(Math.sin(kx), t_1);
	double t_7 = Math.abs(ky) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(ky), 2.0)));
	double tmp;
	if (t_5 <= -0.9948) {
		tmp = t_3;
	} else if (t_5 <= -0.3) {
		tmp = t_6;
	} else if (t_5 <= 5e-15) {
		tmp = (t_1 / Math.sqrt(t_4)) * Math.sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_6;
	} else if (t_5 <= 1.0) {
		tmp = t_3;
	} else {
		tmp = (t_7 / Math.hypot(t_7, Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.pow(t_1, 2.0)
	t_3 = (t_1 / math.sqrt(t_2)) * math.sin(th)
	t_4 = math.pow(math.sin(kx), 2.0)
	t_5 = t_1 / math.sqrt((t_4 + t_2))
	t_6 = (th * t_1) / math.hypot(math.sin(kx), t_1)
	t_7 = math.fabs(ky) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(ky), 2.0)))
	tmp = 0
	if t_5 <= -0.9948:
		tmp = t_3
	elif t_5 <= -0.3:
		tmp = t_6
	elif t_5 <= 5e-15:
		tmp = (t_1 / math.sqrt(t_4)) * math.sin(th)
	elif t_5 <= 0.9990377408563715:
		tmp = t_6
	elif t_5 <= 1.0:
		tmp = t_3
	else:
		tmp = (t_7 / math.hypot(t_7, math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = t_1 ^ 2.0
	t_3 = Float64(Float64(t_1 / sqrt(t_2)) * sin(th))
	t_4 = sin(kx) ^ 2.0
	t_5 = Float64(t_1 / sqrt(Float64(t_4 + t_2)))
	t_6 = Float64(Float64(th * t_1) / hypot(sin(kx), t_1))
	t_7 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	tmp = 0.0
	if (t_5 <= -0.9948)
		tmp = t_3;
	elseif (t_5 <= -0.3)
		tmp = t_6;
	elseif (t_5 <= 5e-15)
		tmp = Float64(Float64(t_1 / sqrt(t_4)) * sin(th));
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_6;
	elseif (t_5 <= 1.0)
		tmp = t_3;
	else
		tmp = Float64(Float64(t_7 / hypot(t_7, sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 ^ 2.0;
	t_3 = (t_1 / sqrt(t_2)) * sin(th);
	t_4 = sin(kx) ^ 2.0;
	t_5 = t_1 / sqrt((t_4 + t_2));
	t_6 = (th * t_1) / hypot(sin(kx), t_1);
	t_7 = abs(ky) * (1.0 + (-0.16666666666666666 * (abs(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_5 <= -0.9948)
		tmp = t_3;
	elseif (t_5 <= -0.3)
		tmp = t_6;
	elseif (t_5 <= 5e-15)
		tmp = (t_1 / sqrt(t_4)) * sin(th);
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_6;
	elseif (t_5 <= 1.0)
		tmp = t_3;
	else
		tmp = (t_7 / hypot(t_7, sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 / N[Sqrt[N[(t$95$4 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(th * t$95$1), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = 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, -0.9948], t$95$3, If[LessEqual[t$95$5, -0.3], t$95$6, If[LessEqual[t$95$5, 5e-15], N[(N[(t$95$1 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.9990377408563715], t$95$6, If[LessEqual[t$95$5, 1.0], t$95$3, N[(N[(t$95$7 / N[Sqrt[t$95$7 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99480000000000002 or 0.99903774085637154 < (/.f64 (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

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6%

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

   4. Applied rewrites 40.6%

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

   if -0.99480000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 96.1%

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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 41.7%

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

   4. Applied rewrites 41.7%

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

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.3%

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.8%

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

   9. Applied rewrites 54.8%

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

Alternative 8: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := {t\_1}^{2}\\ t_3 := \frac{t\_1}{\sqrt{t\_2}} \cdot \sin th\\ t_4 := {\sin kx}^{2}\\ t_5 := \frac{t\_1}{\sqrt{t\_4 + t\_2}}\\ t_6 := \mathsf{hypot}\left(\sin kx, t\_1\right)\\ t_7 := \frac{th \cdot t\_1}{t\_6}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -0.9948:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq -0.3:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_4}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.9990377408563715:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_5 \leq 1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \left|ky\right|\right) \cdot \left(\frac{-1}{t\_6} \cdot \sin th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (pow t_1 2.0))
       (t_3 (* (/ t_1 (sqrt t_2)) (sin th)))
       (t_4 (pow (sin kx) 2.0))
       (t_5 (/ t_1 (sqrt (+ t_4 t_2))))
       (t_6 (hypot (sin kx) t_1))
       (t_7 (/ (* th t_1) t_6)))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -0.9948)
     t_3
     (if (<= t_5 -0.3)
       t_7
       (if (<= t_5 5e-15)
         (* (/ t_1 (sqrt t_4)) (sin th))
         (if (<= t_5 0.9990377408563715)
           t_7
           (if (<= t_5 1.0)
             t_3
             (* (* -1.0 (fabs ky)) (* (/ -1.0 t_6) (sin th)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = pow(t_1, 2.0);
	double t_3 = (t_1 / sqrt(t_2)) * sin(th);
	double t_4 = pow(sin(kx), 2.0);
	double t_5 = t_1 / sqrt((t_4 + t_2));
	double t_6 = hypot(sin(kx), t_1);
	double t_7 = (th * t_1) / t_6;
	double tmp;
	if (t_5 <= -0.9948) {
		tmp = t_3;
	} else if (t_5 <= -0.3) {
		tmp = t_7;
	} else if (t_5 <= 5e-15) {
		tmp = (t_1 / sqrt(t_4)) * sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_7;
	} else if (t_5 <= 1.0) {
		tmp = t_3;
	} else {
		tmp = (-1.0 * fabs(ky)) * ((-1.0 / t_6) * sin(th));
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = Math.pow(t_1, 2.0);
	double t_3 = (t_1 / Math.sqrt(t_2)) * Math.sin(th);
	double t_4 = Math.pow(Math.sin(kx), 2.0);
	double t_5 = t_1 / Math.sqrt((t_4 + t_2));
	double t_6 = Math.hypot(Math.sin(kx), t_1);
	double t_7 = (th * t_1) / t_6;
	double tmp;
	if (t_5 <= -0.9948) {
		tmp = t_3;
	} else if (t_5 <= -0.3) {
		tmp = t_7;
	} else if (t_5 <= 5e-15) {
		tmp = (t_1 / Math.sqrt(t_4)) * Math.sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_7;
	} else if (t_5 <= 1.0) {
		tmp = t_3;
	} else {
		tmp = (-1.0 * Math.abs(ky)) * ((-1.0 / t_6) * Math.sin(th));
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.pow(t_1, 2.0)
	t_3 = (t_1 / math.sqrt(t_2)) * math.sin(th)
	t_4 = math.pow(math.sin(kx), 2.0)
	t_5 = t_1 / math.sqrt((t_4 + t_2))
	t_6 = math.hypot(math.sin(kx), t_1)
	t_7 = (th * t_1) / t_6
	tmp = 0
	if t_5 <= -0.9948:
		tmp = t_3
	elif t_5 <= -0.3:
		tmp = t_7
	elif t_5 <= 5e-15:
		tmp = (t_1 / math.sqrt(t_4)) * math.sin(th)
	elif t_5 <= 0.9990377408563715:
		tmp = t_7
	elif t_5 <= 1.0:
		tmp = t_3
	else:
		tmp = (-1.0 * math.fabs(ky)) * ((-1.0 / t_6) * math.sin(th))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = t_1 ^ 2.0
	t_3 = Float64(Float64(t_1 / sqrt(t_2)) * sin(th))
	t_4 = sin(kx) ^ 2.0
	t_5 = Float64(t_1 / sqrt(Float64(t_4 + t_2)))
	t_6 = hypot(sin(kx), t_1)
	t_7 = Float64(Float64(th * t_1) / t_6)
	tmp = 0.0
	if (t_5 <= -0.9948)
		tmp = t_3;
	elseif (t_5 <= -0.3)
		tmp = t_7;
	elseif (t_5 <= 5e-15)
		tmp = Float64(Float64(t_1 / sqrt(t_4)) * sin(th));
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_7;
	elseif (t_5 <= 1.0)
		tmp = t_3;
	else
		tmp = Float64(Float64(-1.0 * abs(ky)) * Float64(Float64(-1.0 / t_6) * sin(th)));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 ^ 2.0;
	t_3 = (t_1 / sqrt(t_2)) * sin(th);
	t_4 = sin(kx) ^ 2.0;
	t_5 = t_1 / sqrt((t_4 + t_2));
	t_6 = hypot(sin(kx), t_1);
	t_7 = (th * t_1) / t_6;
	tmp = 0.0;
	if (t_5 <= -0.9948)
		tmp = t_3;
	elseif (t_5 <= -0.3)
		tmp = t_7;
	elseif (t_5 <= 5e-15)
		tmp = (t_1 / sqrt(t_4)) * sin(th);
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_7;
	elseif (t_5 <= 1.0)
		tmp = t_3;
	else
		tmp = (-1.0 * abs(ky)) * ((-1.0 / t_6) * sin(th));
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 / N[Sqrt[N[(t$95$4 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]}, Block[{t$95$7 = N[(N[(th * t$95$1), $MachinePrecision] / t$95$6), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, -0.9948], t$95$3, If[LessEqual[t$95$5, -0.3], t$95$7, If[LessEqual[t$95$5, 5e-15], N[(N[(t$95$1 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.9990377408563715], t$95$7, If[LessEqual[t$95$5, 1.0], t$95$3, N[(N[(-1.0 * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / t$95$6), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99480000000000002 or 0.99903774085637154 < (/.f64 (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

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6%

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

   4. Applied rewrites 40.6%

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

   if -0.99480000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 96.1%

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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 41.7%

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

   4. Applied rewrites 41.7%

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

   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. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 51.5%

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

   6. Applied rewrites 51.5%

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

Alternative 9: 82.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := {t\_1}^{2}\\ t_3 := \frac{t\_1}{\sqrt{t\_2}} \cdot \sin th\\ t_4 := \sin \left(\left|kx\right|\right)\\ t_5 := \frac{th \cdot t\_1}{\mathsf{hypot}\left(t\_4, t\_1\right)}\\ t_6 := {t\_4}^{2}\\ t_7 := \frac{t\_1}{\sqrt{t\_6 + t\_2}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_7 \leq -0.9948:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_7 \leq -0.3:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_7 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_6}} \cdot \sin th\\ \mathbf{elif}\;t\_7 \leq 0.9990377408563715:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_7 \leq 2:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\left|ky\right| \cdot \frac{1}{\left|kx\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (pow t_1 2.0))
       (t_3 (* (/ t_1 (sqrt t_2)) (sin th)))
       (t_4 (sin (fabs kx)))
       (t_5 (/ (* th t_1) (hypot t_4 t_1)))
       (t_6 (pow t_4 2.0))
       (t_7 (/ t_1 (sqrt (+ t_6 t_2)))))
  (*
   (copysign 1.0 ky)
   (if (<= t_7 -0.9948)
     t_3
     (if (<= t_7 -0.3)
       t_5
       (if (<= t_7 5e-15)
         (* (/ t_1 (sqrt t_6)) (sin th))
         (if (<= t_7 0.9990377408563715)
           t_5
           (if (<= t_7 2.0)
             t_3
             (* (sin th) (* (fabs ky) (/ 1.0 (fabs kx))))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = pow(t_1, 2.0);
	double t_3 = (t_1 / sqrt(t_2)) * sin(th);
	double t_4 = sin(fabs(kx));
	double t_5 = (th * t_1) / hypot(t_4, t_1);
	double t_6 = pow(t_4, 2.0);
	double t_7 = t_1 / sqrt((t_6 + t_2));
	double tmp;
	if (t_7 <= -0.9948) {
		tmp = t_3;
	} else if (t_7 <= -0.3) {
		tmp = t_5;
	} else if (t_7 <= 5e-15) {
		tmp = (t_1 / sqrt(t_6)) * sin(th);
	} else if (t_7 <= 0.9990377408563715) {
		tmp = t_5;
	} else if (t_7 <= 2.0) {
		tmp = t_3;
	} else {
		tmp = sin(th) * (fabs(ky) * (1.0 / fabs(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.pow(t_1, 2.0);
	double t_3 = (t_1 / Math.sqrt(t_2)) * Math.sin(th);
	double t_4 = Math.sin(Math.abs(kx));
	double t_5 = (th * t_1) / Math.hypot(t_4, t_1);
	double t_6 = Math.pow(t_4, 2.0);
	double t_7 = t_1 / Math.sqrt((t_6 + t_2));
	double tmp;
	if (t_7 <= -0.9948) {
		tmp = t_3;
	} else if (t_7 <= -0.3) {
		tmp = t_5;
	} else if (t_7 <= 5e-15) {
		tmp = (t_1 / Math.sqrt(t_6)) * Math.sin(th);
	} else if (t_7 <= 0.9990377408563715) {
		tmp = t_5;
	} else if (t_7 <= 2.0) {
		tmp = t_3;
	} else {
		tmp = Math.sin(th) * (Math.abs(ky) * (1.0 / Math.abs(kx)));
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.pow(t_1, 2.0)
	t_3 = (t_1 / math.sqrt(t_2)) * math.sin(th)
	t_4 = math.sin(math.fabs(kx))
	t_5 = (th * t_1) / math.hypot(t_4, t_1)
	t_6 = math.pow(t_4, 2.0)
	t_7 = t_1 / math.sqrt((t_6 + t_2))
	tmp = 0
	if t_7 <= -0.9948:
		tmp = t_3
	elif t_7 <= -0.3:
		tmp = t_5
	elif t_7 <= 5e-15:
		tmp = (t_1 / math.sqrt(t_6)) * math.sin(th)
	elif t_7 <= 0.9990377408563715:
		tmp = t_5
	elif t_7 <= 2.0:
		tmp = t_3
	else:
		tmp = math.sin(th) * (math.fabs(ky) * (1.0 / math.fabs(kx)))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = t_1 ^ 2.0
	t_3 = Float64(Float64(t_1 / sqrt(t_2)) * sin(th))
	t_4 = sin(abs(kx))
	t_5 = Float64(Float64(th * t_1) / hypot(t_4, t_1))
	t_6 = t_4 ^ 2.0
	t_7 = Float64(t_1 / sqrt(Float64(t_6 + t_2)))
	tmp = 0.0
	if (t_7 <= -0.9948)
		tmp = t_3;
	elseif (t_7 <= -0.3)
		tmp = t_5;
	elseif (t_7 <= 5e-15)
		tmp = Float64(Float64(t_1 / sqrt(t_6)) * sin(th));
	elseif (t_7 <= 0.9990377408563715)
		tmp = t_5;
	elseif (t_7 <= 2.0)
		tmp = t_3;
	else
		tmp = Float64(sin(th) * Float64(abs(ky) * Float64(1.0 / abs(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 = t_1 ^ 2.0;
	t_3 = (t_1 / sqrt(t_2)) * sin(th);
	t_4 = sin(abs(kx));
	t_5 = (th * t_1) / hypot(t_4, t_1);
	t_6 = t_4 ^ 2.0;
	t_7 = t_1 / sqrt((t_6 + t_2));
	tmp = 0.0;
	if (t_7 <= -0.9948)
		tmp = t_3;
	elseif (t_7 <= -0.3)
		tmp = t_5;
	elseif (t_7 <= 5e-15)
		tmp = (t_1 / sqrt(t_6)) * sin(th);
	elseif (t_7 <= 0.9990377408563715)
		tmp = t_5;
	elseif (t_7 <= 2.0)
		tmp = t_3;
	else
		tmp = sin(th) * (abs(ky) * (1.0 / abs(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[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(th * t$95$1), $MachinePrecision] / N[Sqrt[t$95$4 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$4, 2.0], $MachinePrecision]}, Block[{t$95$7 = N[(t$95$1 / N[Sqrt[N[(t$95$6 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$7, -0.9948], t$95$3, If[LessEqual[t$95$7, -0.3], t$95$5, If[LessEqual[t$95$7, 5e-15], N[(N[(t$95$1 / N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 0.9990377408563715], t$95$5, If[LessEqual[t$95$7, 2.0], t$95$3, N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] * N[(1.0 / N[Abs[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99480000000000002 or 0.99903774085637154 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6%

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

   4. Applied rewrites 40.6%

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

   if -0.99480000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 96.1%

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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 41.7%

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

   4. Applied rewrites 41.7%

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

   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 35.9%

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

   4. Applied rewrites 35.9%

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

   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%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 16.8%

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

   9. Applied rewrites 16.8%

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

Alternative 10: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{th \cdot t\_1}{\mathsf{hypot}\left(\sin kx, t\_1\right)}\\ t_3 := t\_1 \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(\left|ky\right| + \left|ky\right|\right), 0.5, kx \cdot kx\right)}}\\ t_4 := {\sin kx}^{2}\\ t_5 := \frac{t\_1}{\sqrt{t\_4 + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -0.9998:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq -0.3:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_4}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.9990377408563715:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\left|ky\right| \cdot \frac{1}{\left|\sin kx\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (/ (* th t_1) (hypot (sin kx) t_1)))
       (t_3
        (*
         t_1
         (/
          (sin th)
          (sqrt
           (fma
            (- 1.0 (cos (+ (fabs ky) (fabs ky))))
            0.5
            (* kx kx))))))
       (t_4 (pow (sin kx) 2.0))
       (t_5 (/ t_1 (sqrt (+ t_4 (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_5 -0.9998)
     t_3
     (if (<= t_5 -0.3)
       t_2
       (if (<= t_5 5e-15)
         (* (/ t_1 (sqrt t_4)) (sin th))
         (if (<= t_5 0.9990377408563715)
           t_2
           (if (<= t_5 1.0)
             t_3
             (* (sin th) (* (fabs ky) (/ 1.0 (fabs (sin kx)))))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = (th * t_1) / hypot(sin(kx), t_1);
	double t_3 = t_1 * (sin(th) / sqrt(fma((1.0 - cos((fabs(ky) + fabs(ky)))), 0.5, (kx * kx))));
	double t_4 = pow(sin(kx), 2.0);
	double t_5 = t_1 / sqrt((t_4 + pow(t_1, 2.0)));
	double tmp;
	if (t_5 <= -0.9998) {
		tmp = t_3;
	} else if (t_5 <= -0.3) {
		tmp = t_2;
	} else if (t_5 <= 5e-15) {
		tmp = (t_1 / sqrt(t_4)) * sin(th);
	} else if (t_5 <= 0.9990377408563715) {
		tmp = t_2;
	} else if (t_5 <= 1.0) {
		tmp = t_3;
	} else {
		tmp = sin(th) * (fabs(ky) * (1.0 / fabs(sin(kx))));
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(th * t_1) / hypot(sin(kx), t_1))
	t_3 = Float64(t_1 * Float64(sin(th) / sqrt(fma(Float64(1.0 - cos(Float64(abs(ky) + abs(ky)))), 0.5, Float64(kx * kx)))))
	t_4 = sin(kx) ^ 2.0
	t_5 = Float64(t_1 / sqrt(Float64(t_4 + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_5 <= -0.9998)
		tmp = t_3;
	elseif (t_5 <= -0.3)
		tmp = t_2;
	elseif (t_5 <= 5e-15)
		tmp = Float64(Float64(t_1 / sqrt(t_4)) * sin(th));
	elseif (t_5 <= 0.9990377408563715)
		tmp = t_2;
	elseif (t_5 <= 1.0)
		tmp = t_3;
	else
		tmp = Float64(sin(th) * Float64(abs(ky) * Float64(1.0 / 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[(th * t$95$1), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(N[Abs[ky], $MachinePrecision] + N[Abs[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 / N[Sqrt[N[(t$95$4 + 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$5, -0.9998], t$95$3, If[LessEqual[t$95$5, -0.3], t$95$2, If[LessEqual[t$95$5, 5e-15], N[(N[(t$95$1 / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.9990377408563715], t$95$2, If[LessEqual[t$95$5, 1.0], t$95$3, N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] * N[(1.0 / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99980000000000002 or 0.99903774085637154 < (/.f64 (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 52.2%

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

   4. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 52.1%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

   6. Applied rewrites 43.4%

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

   if -0.99980000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 96.1%

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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 41.7%

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

   4. Applied rewrites 41.7%

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

   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. 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 35.9%

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

   4. Applied rewrites 35.9%

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

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 36.7%

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

      11. lift-sqrt.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

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

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

      15. lower-fabs.f64 39.9%

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

   6. Applied rewrites 39.9%

      \[\leadsto \sin th \cdot \color{blue}{\left(ky \cdot \frac{1}{\left|\sin kx\right|}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{th \cdot t\_1}{\mathsf{hypot}\left(\sin kx, t\_1\right)}\\ t_3 := t\_1 \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(\left|ky\right| + \left|ky\right|\right), 0.5, kx \cdot kx\right)}}\\ t_4 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_5 := \left|\sin kx\right|\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.9998:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -0.3:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{-1}{t\_5} \cdot \left(-t\_1\right)\right) \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9990377408563715:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\left|ky\right| \cdot \frac{1}{t\_5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (/ (* th t_1) (hypot (sin kx) t_1)))
       (t_3
        (*
         t_1
         (/
          (sin th)
          (sqrt
           (fma
            (- 1.0 (cos (+ (fabs ky) (fabs ky))))
            0.5
            (* kx kx))))))
       (t_4 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
       (t_5 (fabs (sin kx))))
  (*
   (copysign 1.0 ky)
   (if (<= t_4 -0.9998)
     t_3
     (if (<= t_4 -0.3)
       t_2
       (if (<= t_4 5e-15)
         (* (* (/ -1.0 t_5) (- t_1)) (sin th))
         (if (<= t_4 0.9990377408563715)
           t_2
           (if (<= t_4 1.0)
             t_3
             (* (sin th) (* (fabs ky) (/ 1.0 t_5)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = (th * t_1) / hypot(sin(kx), t_1);
	double t_3 = t_1 * (sin(th) / sqrt(fma((1.0 - cos((fabs(ky) + fabs(ky)))), 0.5, (kx * kx))));
	double t_4 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_5 = fabs(sin(kx));
	double tmp;
	if (t_4 <= -0.9998) {
		tmp = t_3;
	} else if (t_4 <= -0.3) {
		tmp = t_2;
	} else if (t_4 <= 5e-15) {
		tmp = ((-1.0 / t_5) * -t_1) * sin(th);
	} else if (t_4 <= 0.9990377408563715) {
		tmp = t_2;
	} else if (t_4 <= 1.0) {
		tmp = t_3;
	} else {
		tmp = sin(th) * (fabs(ky) * (1.0 / t_5));
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(th * t_1) / hypot(sin(kx), t_1))
	t_3 = Float64(t_1 * Float64(sin(th) / sqrt(fma(Float64(1.0 - cos(Float64(abs(ky) + abs(ky)))), 0.5, Float64(kx * kx)))))
	t_4 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_5 = abs(sin(kx))
	tmp = 0.0
	if (t_4 <= -0.9998)
		tmp = t_3;
	elseif (t_4 <= -0.3)
		tmp = t_2;
	elseif (t_4 <= 5e-15)
		tmp = Float64(Float64(Float64(-1.0 / t_5) * Float64(-t_1)) * sin(th));
	elseif (t_4 <= 0.9990377408563715)
		tmp = t_2;
	elseif (t_4 <= 1.0)
		tmp = t_3;
	else
		tmp = Float64(sin(th) * Float64(abs(ky) * Float64(1.0 / t_5)));
	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[(th * t$95$1), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(N[Abs[ky], $MachinePrecision] + N[Abs[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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$5 = N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, -0.9998], t$95$3, If[LessEqual[t$95$4, -0.3], t$95$2, If[LessEqual[t$95$4, 5e-15], N[(N[(N[(-1.0 / t$95$5), $MachinePrecision] * (-t$95$1)), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9990377408563715], t$95$2, If[LessEqual[t$95$4, 1.0], t$95$3, N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] * N[(1.0 / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99980000000000002 or 0.99903774085637154 < (/.f64 (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 52.2%

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

   4. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 52.1%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

   6. Applied rewrites 43.4%

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

   if -0.99980000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 96.1%

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 41.7%

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

   6. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 45.0%

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

   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. 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 35.9%

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

   4. Applied rewrites 35.9%

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

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 36.7%

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

      11. lift-sqrt.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

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

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

      15. lower-fabs.f64 39.9%

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

   6. Applied rewrites 39.9%

      \[\leadsto \sin th \cdot \color{blue}{\left(ky \cdot \frac{1}{\left|\sin kx\right|}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{th \cdot t\_1}{\mathsf{hypot}\left(\sin kx, t\_1\right)}\\ t_3 := \frac{\sin th \cdot t\_1}{\sqrt{\mathsf{fma}\left(1 - \cos \left(\left|ky\right| + \left|ky\right|\right), 0.5, kx \cdot kx\right)}}\\ t_4 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_5 := \left|\sin kx\right|\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.9998:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -0.3:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{-1}{t\_5} \cdot \left(-t\_1\right)\right) \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9990377408563715:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\left|ky\right| \cdot \frac{1}{t\_5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (/ (* th t_1) (hypot (sin kx) t_1)))
       (t_3
        (/
         (* (sin th) t_1)
         (sqrt
          (fma (- 1.0 (cos (+ (fabs ky) (fabs ky)))) 0.5 (* kx kx)))))
       (t_4 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
       (t_5 (fabs (sin kx))))
  (*
   (copysign 1.0 ky)
   (if (<= t_4 -0.9998)
     t_3
     (if (<= t_4 -0.3)
       t_2
       (if (<= t_4 5e-15)
         (* (* (/ -1.0 t_5) (- t_1)) (sin th))
         (if (<= t_4 0.9990377408563715)
           t_2
           (if (<= t_4 1.0)
             t_3
             (* (sin th) (* (fabs ky) (/ 1.0 t_5)))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = (th * t_1) / hypot(sin(kx), t_1);
	double t_3 = (sin(th) * t_1) / sqrt(fma((1.0 - cos((fabs(ky) + fabs(ky)))), 0.5, (kx * kx)));
	double t_4 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_5 = fabs(sin(kx));
	double tmp;
	if (t_4 <= -0.9998) {
		tmp = t_3;
	} else if (t_4 <= -0.3) {
		tmp = t_2;
	} else if (t_4 <= 5e-15) {
		tmp = ((-1.0 / t_5) * -t_1) * sin(th);
	} else if (t_4 <= 0.9990377408563715) {
		tmp = t_2;
	} else if (t_4 <= 1.0) {
		tmp = t_3;
	} else {
		tmp = sin(th) * (fabs(ky) * (1.0 / t_5));
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(th * t_1) / hypot(sin(kx), t_1))
	t_3 = Float64(Float64(sin(th) * t_1) / sqrt(fma(Float64(1.0 - cos(Float64(abs(ky) + abs(ky)))), 0.5, Float64(kx * kx))))
	t_4 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_5 = abs(sin(kx))
	tmp = 0.0
	if (t_4 <= -0.9998)
		tmp = t_3;
	elseif (t_4 <= -0.3)
		tmp = t_2;
	elseif (t_4 <= 5e-15)
		tmp = Float64(Float64(Float64(-1.0 / t_5) * Float64(-t_1)) * sin(th));
	elseif (t_4 <= 0.9990377408563715)
		tmp = t_2;
	elseif (t_4 <= 1.0)
		tmp = t_3;
	else
		tmp = Float64(sin(th) * Float64(abs(ky) * Float64(1.0 / t_5)));
	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[(th * t$95$1), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[th], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(N[Abs[ky], $MachinePrecision] + N[Abs[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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$5 = N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, -0.9998], t$95$3, If[LessEqual[t$95$4, -0.3], t$95$2, If[LessEqual[t$95$4, 5e-15], N[(N[(N[(-1.0 / t$95$5), $MachinePrecision] * (-t$95$1)), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9990377408563715], t$95$2, If[LessEqual[t$95$4, 1.0], t$95$3, N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] * N[(1.0 / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99980000000000002 or 0.99903774085637154 < (/.f64 (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 52.2%

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

   4. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 50.5%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

   6. Applied rewrites 42.6%

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

   if -0.99980000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.29999999999999999 or 5e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99903774085637154

   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. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 96.1%

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if -0.29999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5e-15

   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. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 41.7%

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

   6. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 45.0%

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

   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. 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 35.9%

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

   4. Applied rewrites 35.9%

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

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 36.7%

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

      11. lift-sqrt.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

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

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

      15. lower-fabs.f64 39.9%

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

   6. Applied rewrites 39.9%

      \[\leadsto \sin th \cdot \color{blue}{\left(ky \cdot \frac{1}{\left|\sin kx\right|}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 68.3% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 31:\\ \;\;\;\;\frac{\left|th\right| \cdot \sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{\left|\sin kx\right|} \cdot \left(-\sin ky\right)\right) \cdot \sin \left(\left|th\right|\right)\\ \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (*
 (copysign 1.0 th)
 (if (<= (fabs th) 31.0)
   (/ (* (fabs th) (sin ky)) (hypot (sin kx) (sin ky)))
   (* (* (/ -1.0 (fabs (sin kx))) (- (sin ky))) (sin (fabs th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 31.0) {
		tmp = (fabs(th) * sin(ky)) / hypot(sin(kx), sin(ky));
	} else {
		tmp = ((-1.0 / fabs(sin(kx))) * -sin(ky)) * sin(fabs(th));
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 31.0) {
		tmp = (Math.abs(th) * Math.sin(ky)) / Math.hypot(Math.sin(kx), Math.sin(ky));
	} else {
		tmp = ((-1.0 / Math.abs(Math.sin(kx))) * -Math.sin(ky)) * Math.sin(Math.abs(th));
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 31.0:
		tmp = (math.fabs(th) * math.sin(ky)) / math.hypot(math.sin(kx), math.sin(ky))
	else:
		tmp = ((-1.0 / math.fabs(math.sin(kx))) * -math.sin(ky)) * math.sin(math.fabs(th))
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 31.0)
		tmp = Float64(Float64(abs(th) * sin(ky)) / hypot(sin(kx), sin(ky)));
	else
		tmp = Float64(Float64(Float64(-1.0 / abs(sin(kx))) * Float64(-sin(ky))) * sin(abs(th)));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 31.0)
		tmp = (abs(th) * sin(ky)) / hypot(sin(kx), sin(ky));
	else
		tmp = ((-1.0 / abs(sin(kx))) * -sin(ky)) * sin(abs(th));
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 31.0], N[(N[(N[Abs[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sin[ky], $MachinePrecision])), $MachinePrecision] * N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 31

   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. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 96.1%

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 47.6%

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

   6. Applied rewrites 47.6%

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

   if 31 < th

   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. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 41.7%

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

   6. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 45.0%

      \[\leadsto \color{blue}{\left(\frac{-1}{\left|\sin kx\right|} \cdot \left(-\sin ky\right)\right) \cdot \sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 53.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (sin (fabs kx)))
       (t_3 (/ t_1 (sqrt (+ (pow t_2 2.0) (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_3 0.99)
     (* (* (/ -1.0 (fabs t_2)) (- t_1)) (sin th))
     (if (<= t_3 2.0)
       (*
        (/
         t_1
         (sqrt (+ (* (fabs ky) (fabs ky)) (* (fabs kx) (fabs kx)))))
        (sin th))
       (* (sin th) (* (fabs ky) (/ 1.0 (fabs kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(fabs(kx));
	double t_3 = t_1 / sqrt((pow(t_2, 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.99) {
		tmp = ((-1.0 / fabs(t_2)) * -t_1) * sin(th);
	} else if (t_3 <= 2.0) {
		tmp = (t_1 / sqrt(((fabs(ky) * fabs(ky)) + (fabs(kx) * fabs(kx))))) * sin(th);
	} else {
		tmp = sin(th) * (fabs(ky) * (1.0 / fabs(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(Math.abs(kx));
	double t_3 = t_1 / Math.sqrt((Math.pow(t_2, 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.99) {
		tmp = ((-1.0 / Math.abs(t_2)) * -t_1) * Math.sin(th);
	} else if (t_3 <= 2.0) {
		tmp = (t_1 / Math.sqrt(((Math.abs(ky) * Math.abs(ky)) + (Math.abs(kx) * Math.abs(kx))))) * Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.abs(ky) * (1.0 / Math.abs(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(math.fabs(kx))
	t_3 = t_1 / math.sqrt((math.pow(t_2, 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_3 <= 0.99:
		tmp = ((-1.0 / math.fabs(t_2)) * -t_1) * math.sin(th)
	elif t_3 <= 2.0:
		tmp = (t_1 / math.sqrt(((math.fabs(ky) * math.fabs(ky)) + (math.fabs(kx) * math.fabs(kx))))) * math.sin(th)
	else:
		tmp = math.sin(th) * (math.fabs(ky) * (1.0 / math.fabs(kx)))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(abs(kx))
	t_3 = Float64(t_1 / sqrt(Float64((t_2 ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 0.99)
		tmp = Float64(Float64(Float64(-1.0 / abs(t_2)) * Float64(-t_1)) * sin(th));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(t_1 / sqrt(Float64(Float64(abs(ky) * abs(ky)) + Float64(abs(kx) * abs(kx))))) * sin(th));
	else
		tmp = Float64(sin(th) * Float64(abs(ky) * Float64(1.0 / abs(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(abs(kx));
	t_3 = t_1 / sqrt(((t_2 ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= 0.99)
		tmp = ((-1.0 / abs(t_2)) * -t_1) * sin(th);
	elseif (t_3 <= 2.0)
		tmp = (t_1 / sqrt(((abs(ky) * abs(ky)) + (abs(kx) * abs(kx))))) * sin(th);
	else
		tmp = sin(th) * (abs(ky) * (1.0 / abs(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[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$2, 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, 0.99], N[(N[(N[(-1.0 / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision] * (-t$95$1)), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(t$95$1 / N[Sqrt[N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[kx], $MachinePrecision] * N[Abs[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] * N[(1.0 / N[Abs[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))))) < 0.98999999999999999

   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. frac-2neg N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 41.7%

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

   6. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 45.0%

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

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

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 52.2%

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in ky around 0

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

      1. lower-pow.f64 28.4%

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

   7. Applied rewrites 28.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 28.4%

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 28.4%

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

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{ky \cdot \color{blue}{ky} + \mathsf{Rewrite=>}\left(lift-pow.f64, \left({kx}^{2}\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{ky \cdot \color{blue}{ky} + \mathsf{Rewrite=>}\left(unpow2, \left(kx \cdot kx\right)\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{ky \cdot \color{blue}{ky} + \mathsf{Rewrite=>}\left(lower-*.f64, \left(kx \cdot kx\right)\right)}} \cdot \sin th \]

   9. Applied rewrites 28.4%

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

   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 35.9%

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

   4. Applied rewrites 35.9%

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

   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%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 16.8%

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

   9. Applied rewrites 16.8%

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

Alternative 15: 48.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \sin \left(\left|kx\right|\right)\\ t_3 := \frac{t\_1}{\sqrt{{t\_2}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 0.965:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|t\_2\right|}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{t\_1}{\sqrt{\left|ky\right| \cdot \left|ky\right| + \left|kx\right| \cdot \left|kx\right|}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\left|ky\right| \cdot \frac{1}{\left|kx\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (sin (fabs kx)))
       (t_3 (/ t_1 (sqrt (+ (pow t_2 2.0) (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_3 0.965)
     (* (sin th) (/ (fabs ky) (fabs t_2)))
     (if (<= t_3 2.0)
       (*
        (/
         t_1
         (sqrt (+ (* (fabs ky) (fabs ky)) (* (fabs kx) (fabs kx)))))
        (sin th))
       (* (sin th) (* (fabs ky) (/ 1.0 (fabs kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(fabs(kx));
	double t_3 = t_1 / sqrt((pow(t_2, 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.965) {
		tmp = sin(th) * (fabs(ky) / fabs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = (t_1 / sqrt(((fabs(ky) * fabs(ky)) + (fabs(kx) * fabs(kx))))) * sin(th);
	} else {
		tmp = sin(th) * (fabs(ky) * (1.0 / fabs(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(Math.abs(kx));
	double t_3 = t_1 / Math.sqrt((Math.pow(t_2, 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.965) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = (t_1 / Math.sqrt(((Math.abs(ky) * Math.abs(ky)) + (Math.abs(kx) * Math.abs(kx))))) * Math.sin(th);
	} else {
		tmp = Math.sin(th) * (Math.abs(ky) * (1.0 / Math.abs(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(math.fabs(kx))
	t_3 = t_1 / math.sqrt((math.pow(t_2, 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_3 <= 0.965:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(t_2))
	elif t_3 <= 2.0:
		tmp = (t_1 / math.sqrt(((math.fabs(ky) * math.fabs(ky)) + (math.fabs(kx) * math.fabs(kx))))) * math.sin(th)
	else:
		tmp = math.sin(th) * (math.fabs(ky) * (1.0 / math.fabs(kx)))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(abs(kx))
	t_3 = Float64(t_1 / sqrt(Float64((t_2 ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 0.965)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(t_2)));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(t_1 / sqrt(Float64(Float64(abs(ky) * abs(ky)) + Float64(abs(kx) * abs(kx))))) * sin(th));
	else
		tmp = Float64(sin(th) * Float64(abs(ky) * Float64(1.0 / abs(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(abs(kx));
	t_3 = t_1 / sqrt(((t_2 ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= 0.965)
		tmp = sin(th) * (abs(ky) / abs(t_2));
	elseif (t_3 <= 2.0)
		tmp = (t_1 / sqrt(((abs(ky) * abs(ky)) + (abs(kx) * abs(kx))))) * sin(th);
	else
		tmp = sin(th) * (abs(ky) * (1.0 / abs(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[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$2, 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, 0.965], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(t$95$1 / N[Sqrt[N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[kx], $MachinePrecision] * N[Abs[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] * N[(1.0 / N[Abs[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))))) < 0.96499999999999997

   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 35.9%

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

   4. Applied rewrites 35.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 36.7%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 40.0%

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

   6. Applied rewrites 40.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 39.9%

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

   8. Applied rewrites 39.9%

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

   if 0.96499999999999997 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 52.2%

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in ky around 0

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

      1. lower-pow.f64 28.4%

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

   7. Applied rewrites 28.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 28.4%

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 28.4%

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

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{ky \cdot \color{blue}{ky} + \mathsf{Rewrite=>}\left(lift-pow.f64, \left({kx}^{2}\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{ky \cdot \color{blue}{ky} + \mathsf{Rewrite=>}\left(unpow2, \left(kx \cdot kx\right)\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{ky \cdot \color{blue}{ky} + \mathsf{Rewrite=>}\left(lower-*.f64, \left(kx \cdot kx\right)\right)}} \cdot \sin th \]

   9. Applied rewrites 28.4%

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

   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 35.9%

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

   4. Applied rewrites 35.9%

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

   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%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 16.8%

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

   9. Applied rewrites 16.8%

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

Alternative 16: 48.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \sin \left(\left|kx\right|\right)\\ t_3 := \frac{t\_1}{\sqrt{{t\_2}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 0.965:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|t\_2\right|}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\sqrt{\left|ky\right| \cdot \left|ky\right| + \left|kx\right| \cdot \left|kx\right|}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\left|ky\right| \cdot \frac{1}{\left|kx\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (sin (fabs kx)))
       (t_3 (/ t_1 (sqrt (+ (pow t_2 2.0) (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_3 0.965)
     (* (sin th) (/ (fabs ky) (fabs t_2)))
     (if (<= t_3 2.0)
       (*
        t_1
        (/
         (sin th)
         (sqrt (+ (* (fabs ky) (fabs ky)) (* (fabs kx) (fabs kx))))))
       (* (sin th) (* (fabs ky) (/ 1.0 (fabs kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(fabs(kx));
	double t_3 = t_1 / sqrt((pow(t_2, 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.965) {
		tmp = sin(th) * (fabs(ky) / fabs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = t_1 * (sin(th) / sqrt(((fabs(ky) * fabs(ky)) + (fabs(kx) * fabs(kx)))));
	} else {
		tmp = sin(th) * (fabs(ky) * (1.0 / fabs(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(Math.abs(kx));
	double t_3 = t_1 / Math.sqrt((Math.pow(t_2, 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.965) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = t_1 * (Math.sin(th) / Math.sqrt(((Math.abs(ky) * Math.abs(ky)) + (Math.abs(kx) * Math.abs(kx)))));
	} else {
		tmp = Math.sin(th) * (Math.abs(ky) * (1.0 / Math.abs(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(math.fabs(kx))
	t_3 = t_1 / math.sqrt((math.pow(t_2, 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_3 <= 0.965:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(t_2))
	elif t_3 <= 2.0:
		tmp = t_1 * (math.sin(th) / math.sqrt(((math.fabs(ky) * math.fabs(ky)) + (math.fabs(kx) * math.fabs(kx)))))
	else:
		tmp = math.sin(th) * (math.fabs(ky) * (1.0 / math.fabs(kx)))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(abs(kx))
	t_3 = Float64(t_1 / sqrt(Float64((t_2 ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 0.965)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(t_2)));
	elseif (t_3 <= 2.0)
		tmp = Float64(t_1 * Float64(sin(th) / sqrt(Float64(Float64(abs(ky) * abs(ky)) + Float64(abs(kx) * abs(kx))))));
	else
		tmp = Float64(sin(th) * Float64(abs(ky) * Float64(1.0 / abs(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(abs(kx));
	t_3 = t_1 / sqrt(((t_2 ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= 0.965)
		tmp = sin(th) * (abs(ky) / abs(t_2));
	elseif (t_3 <= 2.0)
		tmp = t_1 * (sin(th) / sqrt(((abs(ky) * abs(ky)) + (abs(kx) * abs(kx)))));
	else
		tmp = sin(th) * (abs(ky) * (1.0 / abs(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[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$2, 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, 0.965], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[kx], $MachinePrecision] * N[Abs[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] * N[(1.0 / N[Abs[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))))) < 0.96499999999999997

   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 35.9%

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

   4. Applied rewrites 35.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 36.7%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 40.0%

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

   6. Applied rewrites 40.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 39.9%

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

   8. Applied rewrites 39.9%

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

   if 0.96499999999999997 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 52.2%

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in ky around 0

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

      1. lower-pow.f64 28.4%

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

   7. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 28.3%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 28.3%

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

   9. Applied rewrites 28.3%

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

   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 35.9%

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

   4. Applied rewrites 35.9%

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

   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%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 16.8%

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

   9. Applied rewrites 16.8%

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

Alternative 17: 47.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (sin (fabs kx)))
       (t_3 (/ t_1 (sqrt (+ (pow t_2 2.0) (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_3 0.99)
     (* (sin th) (/ (fabs ky) (fabs t_2)))
     (if (<= t_3 2.0)
       (/
        1.0
        (/
         (sqrt (+ (* (fabs ky) (fabs ky)) (* (fabs kx) (fabs kx))))
         (*
          (*
           (fabs ky)
           (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0))))
          (sin th))))
       (* (sin th) (* (fabs ky) (/ 1.0 (fabs kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(fabs(kx));
	double t_3 = t_1 / sqrt((pow(t_2, 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.99) {
		tmp = sin(th) * (fabs(ky) / fabs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = 1.0 / (sqrt(((fabs(ky) * fabs(ky)) + (fabs(kx) * fabs(kx)))) / ((fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)))) * sin(th)));
	} else {
		tmp = sin(th) * (fabs(ky) * (1.0 / fabs(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(Math.abs(kx));
	double t_3 = t_1 / Math.sqrt((Math.pow(t_2, 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.99) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = 1.0 / (Math.sqrt(((Math.abs(ky) * Math.abs(ky)) + (Math.abs(kx) * Math.abs(kx)))) / ((Math.abs(ky) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(ky), 2.0)))) * Math.sin(th)));
	} else {
		tmp = Math.sin(th) * (Math.abs(ky) * (1.0 / Math.abs(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(math.fabs(kx))
	t_3 = t_1 / math.sqrt((math.pow(t_2, 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_3 <= 0.99:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(t_2))
	elif t_3 <= 2.0:
		tmp = 1.0 / (math.sqrt(((math.fabs(ky) * math.fabs(ky)) + (math.fabs(kx) * math.fabs(kx)))) / ((math.fabs(ky) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(ky), 2.0)))) * math.sin(th)))
	else:
		tmp = math.sin(th) * (math.fabs(ky) * (1.0 / math.fabs(kx)))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(abs(kx))
	t_3 = Float64(t_1 / sqrt(Float64((t_2 ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 0.99)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(t_2)));
	elseif (t_3 <= 2.0)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(abs(ky) * abs(ky)) + Float64(abs(kx) * abs(kx)))) / Float64(Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0)))) * sin(th))));
	else
		tmp = Float64(sin(th) * Float64(abs(ky) * Float64(1.0 / abs(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(abs(kx));
	t_3 = t_1 / sqrt(((t_2 ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= 0.99)
		tmp = sin(th) * (abs(ky) / abs(t_2));
	elseif (t_3 <= 2.0)
		tmp = 1.0 / (sqrt(((abs(ky) * abs(ky)) + (abs(kx) * abs(kx)))) / ((abs(ky) * (1.0 + (-0.16666666666666666 * (abs(ky) ^ 2.0)))) * sin(th)));
	else
		tmp = sin(th) * (abs(ky) * (1.0 / abs(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[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$2, 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, 0.99], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(1.0 / N[(N[Sqrt[N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[kx], $MachinePrecision] * N[Abs[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] * N[(1.0 / N[Abs[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))))) < 0.98999999999999999

   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 35.9%

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

   4. Applied rewrites 35.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 36.7%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 40.0%

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

   6. Applied rewrites 40.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 39.9%

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

   8. Applied rewrites 39.9%

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

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

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 52.2%

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in ky around 0

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

      1. lower-pow.f64 28.4%

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

   7. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{ky \cdot ky + kx \cdot kx}}{\sin ky \cdot \sin th}}} \]

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 24.9%

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

   12. Applied rewrites 24.9%

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

   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 35.9%

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

   4. Applied rewrites 35.9%

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

   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%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 16.8%

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

   9. Applied rewrites 16.8%

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

Alternative 18: 43.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (kx ky th)
  :precision binary64
  (let* ((t_1 (sin (fabs ky)))
       (t_2 (sin (fabs kx)))
       (t_3 (/ t_1 (sqrt (+ (pow t_2 2.0) (pow t_1 2.0))))))
  (*
   (copysign 1.0 ky)
   (if (<= t_3 0.99)
     (* (sin th) (/ (fabs ky) (fabs t_2)))
     (if (<= t_3 2.0)
       (/
        1.0
        (/
         (sqrt (+ (* (fabs ky) (fabs ky)) (* (fabs kx) (fabs kx))))
         (* th t_1)))
       (* (sin th) (* (fabs ky) (/ 1.0 (fabs kx)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(fabs(kx));
	double t_3 = t_1 / sqrt((pow(t_2, 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.99) {
		tmp = sin(th) * (fabs(ky) / fabs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = 1.0 / (sqrt(((fabs(ky) * fabs(ky)) + (fabs(kx) * fabs(kx)))) / (th * t_1));
	} else {
		tmp = sin(th) * (fabs(ky) * (1.0 / fabs(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(Math.abs(kx));
	double t_3 = t_1 / Math.sqrt((Math.pow(t_2, 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.99) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = 1.0 / (Math.sqrt(((Math.abs(ky) * Math.abs(ky)) + (Math.abs(kx) * Math.abs(kx)))) / (th * t_1));
	} else {
		tmp = Math.sin(th) * (Math.abs(ky) * (1.0 / Math.abs(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(math.fabs(kx))
	t_3 = t_1 / math.sqrt((math.pow(t_2, 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_3 <= 0.99:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(t_2))
	elif t_3 <= 2.0:
		tmp = 1.0 / (math.sqrt(((math.fabs(ky) * math.fabs(ky)) + (math.fabs(kx) * math.fabs(kx)))) / (th * t_1))
	else:
		tmp = math.sin(th) * (math.fabs(ky) * (1.0 / math.fabs(kx)))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(abs(kx))
	t_3 = Float64(t_1 / sqrt(Float64((t_2 ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 0.99)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(t_2)));
	elseif (t_3 <= 2.0)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(abs(ky) * abs(ky)) + Float64(abs(kx) * abs(kx)))) / Float64(th * t_1)));
	else
		tmp = Float64(sin(th) * Float64(abs(ky) * Float64(1.0 / abs(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(abs(kx));
	t_3 = t_1 / sqrt(((t_2 ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= 0.99)
		tmp = sin(th) * (abs(ky) / abs(t_2));
	elseif (t_3 <= 2.0)
		tmp = 1.0 / (sqrt(((abs(ky) * abs(ky)) + (abs(kx) * abs(kx)))) / (th * t_1));
	else
		tmp = sin(th) * (abs(ky) * (1.0 / abs(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[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$2, 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, 0.99], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(1.0 / N[(N[Sqrt[N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] + N[(N[Abs[kx], $MachinePrecision] * N[Abs[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(th * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] * N[(1.0 / N[Abs[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))))) < 0.98999999999999999

   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 35.9%

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

   4. Applied rewrites 35.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 36.7%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 40.0%

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

   6. Applied rewrites 40.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 39.9%

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

   8. Applied rewrites 39.9%

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

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

   1. Initial program 93.7%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 52.2%

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in ky around 0

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

      1. lower-pow.f64 28.4%

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

   7. Applied rewrites 28.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{ky \cdot ky + kx \cdot kx}}{\sin ky \cdot \sin th}}} \]

   10. Taylor expanded in th around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{ky \cdot ky + kx \cdot kx}}{th \cdot \color{blue}{\sin ky}}} \]

       2. lower-sin.f64 17.4%

          \[\leadsto \frac{1}{\frac{\sqrt{ky \cdot ky + kx \cdot kx}}{th \cdot \sin ky}} \]

   12. Applied rewrites 17.4%

       \[\leadsto \frac{1}{\frac{\sqrt{ky \cdot ky + kx \cdot kx}}{\color{blue}{th \cdot \sin ky}}} \]

   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 35.9%

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

   4. Applied rewrites 35.9%

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

   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%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 16.8%

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

   9. Applied rewrites 16.8%

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

Alternative 19: 27.7% accurate, 3.3× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 420:\\ \;\;\;\;\frac{\left|th\right|}{\left|\sin \left(\left|kx\right|\right)\right|} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|th\right|\right) \cdot \frac{ky}{\left|kx\right|}\\ \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (*
 (copysign 1.0 th)
 (if (<= (fabs th) 420.0)
   (* (/ (fabs th) (fabs (sin (fabs kx)))) ky)
   (* (sin (fabs th)) (/ ky (fabs kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 420.0) {
		tmp = (fabs(th) / fabs(sin(fabs(kx)))) * ky;
	} else {
		tmp = sin(fabs(th)) * (ky / fabs(kx));
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 420.0) {
		tmp = (Math.abs(th) / Math.abs(Math.sin(Math.abs(kx)))) * ky;
	} else {
		tmp = Math.sin(Math.abs(th)) * (ky / Math.abs(kx));
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 420.0:
		tmp = (math.fabs(th) / math.fabs(math.sin(math.fabs(kx)))) * ky
	else:
		tmp = math.sin(math.fabs(th)) * (ky / math.fabs(kx))
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 420.0)
		tmp = Float64(Float64(abs(th) / abs(sin(abs(kx)))) * ky);
	else
		tmp = Float64(sin(abs(th)) * Float64(ky / abs(kx)));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 420.0)
		tmp = (abs(th) / abs(sin(abs(kx)))) * ky;
	else
		tmp = sin(abs(th)) * (ky / abs(kx));
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 420.0], N[(N[(N[Abs[th], $MachinePrecision] / N[Abs[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision], N[(N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision] * N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 420

   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 35.9%

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

   4. Applied rewrites 35.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 36.7%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in th around 0

      \[\leadsto \frac{th}{\left|\sin kx\right|} \cdot ky \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fabs.f64 N/A

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

      3. lower-sin.f64 21.2%

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

   9. Applied rewrites 21.2%

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

   if 420 < th

   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 35.9%

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

   4. Applied rewrites 35.9%

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

   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%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 16.8%

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

   9. Applied rewrites 16.8%

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

Alternative 20: 26.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|kx\right| \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\frac{\sin th}{\left|kx\right|} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot th}{\left|\sin \left(\left|kx\right|\right)\right|}\\ \end{array} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (if (<= (fabs kx) 3e+28)
  (* (/ (sin th) (fabs kx)) ky)
  (/ (* ky th) (fabs (sin (fabs kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(kx) <= 3e+28) {
		tmp = (sin(th) / fabs(kx)) * ky;
	} else {
		tmp = (ky * th) / fabs(sin(fabs(kx)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (abs(kx) <= 3d+28) then
        tmp = (sin(th) / abs(kx)) * ky
    else
        tmp = (ky * th) / abs(sin(abs(kx)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(kx) <= 3e+28) {
		tmp = (Math.sin(th) / Math.abs(kx)) * ky;
	} else {
		tmp = (ky * th) / Math.abs(Math.sin(Math.abs(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(kx) <= 3e+28:
		tmp = (math.sin(th) / math.fabs(kx)) * ky
	else:
		tmp = (ky * th) / math.fabs(math.sin(math.fabs(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(kx) <= 3e+28)
		tmp = Float64(Float64(sin(th) / abs(kx)) * ky);
	else
		tmp = Float64(Float64(ky * th) / abs(sin(abs(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(kx) <= 3e+28)
		tmp = (sin(th) / abs(kx)) * ky;
	else
		tmp = (ky * th) / abs(sin(abs(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Abs[kx], $MachinePrecision], 3e+28], N[(N[(N[Sin[th], $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision], N[(N[(ky * th), $MachinePrecision] / N[Abs[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 3.0000000000000001e28

   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 35.9%

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

   4. Applied rewrites 35.9%

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

   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%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 16.8%

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

   9. Applied rewrites 16.8%

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

   if 3.0000000000000001e28 < kx

   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 35.9%

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

   4. Applied rewrites 35.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 36.7%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 40.0%

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

   6. Applied rewrites 40.0%

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

   7. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sin.f64 19.5%

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

   9. Applied rewrites 19.5%

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

Alternative 21: 22.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(kx, ky, th):
	return (math.sin(th) / math.fabs(kx)) * ky

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(th) / abs(kx)) * ky)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(th) / abs(kx)) * ky;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[th], $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision]

Derivation

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 35.9%

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

4. Applied rewrites 35.9%

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

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%

   \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   3. associate-/l* N/A

      \[\leadsto ky \cdot \frac{\sin th}{\color{blue}{kx}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sin th}{kx} \cdot ky \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\sin th}{kx} \cdot ky \]

   6. lower-/.f64 16.8%

      \[\leadsto \frac{\sin th}{kx} \cdot ky \]

9. Applied rewrites 16.8%

   \[\leadsto \frac{\sin th}{kx} \cdot ky \]
10. Add Preprocessing

Alternative 22: 22.1% accurate, 4.3× speedup

Math

\[\sin th \cdot \frac{ky}{\left|kx\right|} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (* (sin th) (/ ky (fabs kx))))

C

double code(double kx, double ky, double th) {
	return sin(th) * (ky / fabs(kx));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = sin(th) * (ky / abs(kx))
end function

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th) * (ky / Math.abs(kx));
}

Python

def code(kx, ky, th):
	return math.sin(th) * (ky / math.fabs(kx))

Julia

function code(kx, ky, th)
	return Float64(sin(th) * Float64(ky / abs(kx)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th) * (ky / abs(kx));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Abs[kx], $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 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 35.9%

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

4. Applied rewrites 35.9%

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

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%

   \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   3. *-commutative N/A

      \[\leadsto \frac{\sin th \cdot ky}{kx} \]

   4. associate-/l* N/A

      \[\leadsto \sin th \cdot \frac{ky}{\color{blue}{kx}} \]

   5. lower-*.f64 N/A

      \[\leadsto \sin th \cdot \frac{ky}{\color{blue}{kx}} \]

   6. lower-/.f64 16.8%

      \[\leadsto \sin th \cdot \frac{ky}{kx} \]

9. Applied rewrites 16.8%

   \[\leadsto \sin th \cdot \frac{ky}{\color{blue}{kx}} \]
10. Add Preprocessing

Alternative 23: 15.1% accurate, 4.7× speedup

Math

\[th \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{ky \cdot {th}^{2}}{\left|kx\right|}, \frac{ky}{\left|kx\right|}\right) \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (*
 th
 (fma
  -0.16666666666666666
  (/ (* ky (pow th 2.0)) (fabs kx))
  (/ ky (fabs kx)))))

C

double code(double kx, double ky, double th) {
	return th * fma(-0.16666666666666666, ((ky * pow(th, 2.0)) / fabs(kx)), (ky / fabs(kx)));
}

Julia

function code(kx, ky, th)
	return Float64(th * fma(-0.16666666666666666, Float64(Float64(ky * (th ^ 2.0)) / abs(kx)), Float64(ky / abs(kx))))
end

Wolfram

code[kx_, ky_, th_] := N[(th * N[(-0.16666666666666666 * N[(N[(ky * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision] + N[(ky / N[Abs[kx], $MachinePrecision]), $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 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 35.9%

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

4. Applied rewrites 35.9%

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

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%

   \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]

8. Taylor expanded in th around 0

   \[\leadsto th \cdot \left(\frac{-1}{6} \cdot \frac{ky \cdot {th}^{2}}{kx} + \color{blue}{\frac{ky}{kx}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto th \cdot \left(\frac{-1}{6} \cdot \frac{ky \cdot {th}^{2}}{kx} + \frac{ky}{\color{blue}{kx}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{ky \cdot {th}^{2}}{kx}, \frac{ky}{kx}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{ky \cdot {th}^{2}}{kx}, \frac{ky}{kx}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{ky \cdot {th}^{2}}{kx}, \frac{ky}{kx}\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto th \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{ky \cdot {th}^{2}}{kx}, \frac{ky}{kx}\right) \]

   6. lower-/.f64 13.2%

      \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{ky \cdot {th}^{2}}{kx}, \frac{ky}{kx}\right) \]

10. Applied rewrites 13.2%

    \[\leadsto th \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{ky \cdot {th}^{2}}{kx}}, \frac{ky}{kx}\right) \]
11. Add Preprocessing

Alternative 24: 14.0% accurate, 20.5× speedup

Math

\[\frac{ky \cdot th}{\left|kx\right|} \]

FPCore

(FPCore (kx ky th)
  :precision binary64
  (/ (* ky th) (fabs kx)))

C

double code(double kx, double ky, double th) {
	return (ky * th) / fabs(kx);
}

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 = (ky * th) / abs(kx)
end function

Java

public static double code(double kx, double ky, double th) {
	return (ky * th) / Math.abs(kx);
}

Python

def code(kx, ky, th):
	return (ky * th) / math.fabs(kx)

Julia

function code(kx, ky, th)
	return Float64(Float64(ky * th) / abs(kx))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (ky * th) / abs(kx);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(ky * th), $MachinePrecision] / N[Abs[kx], $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 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 35.9%

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

4. Applied rewrites 35.9%

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

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%

   \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]

8. Taylor expanded in th around 0

   \[\leadsto \frac{ky \cdot th}{kx} \]
9. Step-by-step derivation

   1. lower-*.f64 12.9%

      \[\leadsto \frac{ky \cdot th}{kx} \]

10. Applied rewrites 12.9%

    \[\leadsto \frac{ky \cdot th}{kx} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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)))