Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.7%

Time: 6.2s

Alternatives: 20

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: 94.2% 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 94.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7%

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

3. Applied rewrites 99.7%

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

Alternative 2: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \frac{t\_2}{\sqrt{{\sin kx}^{2} + {t\_2}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.99996:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(t\_2, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(t\_2, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 0.9999:\\ \;\;\;\;\frac{\frac{t\_2}{\mathsf{hypot}\left(\sin kx, t\_2\right)}}{\frac{1}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (* (fabs ky) (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0)))))
        (t_2 (sin (fabs ky)))
        (t_3 (/ t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_2 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.99996)
      (* (/ t_2 (hypot t_2 kx)) (sin th))
      (if (<= t_3 -0.1)
        (* (/ t_2 (hypot t_2 (sin kx))) th)
        (if (<= t_3 0.05)
          (* (/ (sin th) (hypot (sin kx) t_1)) t_1)
          (if (<= t_3 0.9999)
            (/ (/ t_2 (hypot (sin kx) t_2)) (/ 1.0 th))
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double t_2 = sin(fabs(ky));
	double t_3 = t_2 / sqrt((pow(sin(kx), 2.0) + pow(t_2, 2.0)));
	double tmp;
	if (t_3 <= -0.99996) {
		tmp = (t_2 / hypot(t_2, kx)) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (t_2 / hypot(t_2, sin(kx))) * th;
	} else if (t_3 <= 0.05) {
		tmp = (sin(th) / hypot(sin(kx), t_1)) * t_1;
	} else if (t_3 <= 0.9999) {
		tmp = (t_2 / hypot(sin(kx), t_2)) / (1.0 / th);
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), 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.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_2, 2.0)));
	double tmp;
	if (t_3 <= -0.99996) {
		tmp = (t_2 / Math.hypot(t_2, kx)) * Math.sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (t_2 / Math.hypot(t_2, Math.sin(kx))) * th;
	} else if (t_3 <= 0.05) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), t_1)) * t_1;
	} else if (t_3 <= 0.9999) {
		tmp = (t_2 / Math.hypot(Math.sin(kx), t_2)) / (1.0 / th);
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), 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.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_2, 2.0)))
	tmp = 0
	if t_3 <= -0.99996:
		tmp = (t_2 / math.hypot(t_2, kx)) * math.sin(th)
	elif t_3 <= -0.1:
		tmp = (t_2 / math.hypot(t_2, math.sin(kx))) * th
	elif t_3 <= 0.05:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), t_1)) * t_1
	elif t_3 <= 0.9999:
		tmp = (t_2 / math.hypot(math.sin(kx), t_2)) / (1.0 / th)
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), 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(t_2 / sqrt(Float64((sin(kx) ^ 2.0) + (t_2 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.99996)
		tmp = Float64(Float64(t_2 / hypot(t_2, kx)) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(t_2 / hypot(t_2, sin(kx))) * th);
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), t_1)) * t_1);
	elseif (t_3 <= 0.9999)
		tmp = Float64(Float64(t_2 / hypot(sin(kx), t_2)) / Float64(1.0 / th));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), 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 / sqrt(((sin(kx) ^ 2.0) + (t_2 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -0.99996)
		tmp = (t_2 / hypot(t_2, kx)) * sin(th);
	elseif (t_3 <= -0.1)
		tmp = (t_2 / hypot(t_2, sin(kx))) * th;
	elseif (t_3 <= 0.05)
		tmp = (sin(th) / hypot(sin(kx), t_1)) * t_1;
	elseif (t_3 <= 0.9999)
		tmp = (t_2 / hypot(sin(kx), t_2)) / (1.0 / th);
	else
		tmp = (abs(ky) / hypot(abs(ky), 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[(t$95$2 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.99996], N[(N[(t$95$2 / N[Sqrt[t$95$2 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], N[(N[(t$95$2 / N[Sqrt[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.9999], N[(N[(t$95$2 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision] / N[(1.0 / th), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 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))))) < -0.99995999999999996

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 kx around 0

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

   6. Applied rewrites 57.6%

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

   if -0.99995999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.1%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.9%

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

   6. Applied rewrites 51.9%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.0%

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

   9. Applied rewrites 54.0%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99990000000000001

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. div-flip N/A

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

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

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

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

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.7%

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 51.0%

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

      8. lift-hypot.f64 N/A

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

      9. pow2 N/A

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

      10. lift-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lower-hypot.f64 51.0%

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

   8. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow1/2 N/A

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

      14. lower-/.f64 N/A

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

   10. Applied rewrites 51.0%

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

   if 0.99990000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.6%

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

Alternative 3: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.99996:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)} \cdot \left|ky\right|\\ \mathbf{elif}\;t\_2 \leq 0.9999:\\ \;\;\;\;\frac{\frac{t\_1}{\mathsf{hypot}\left(\sin kx, t\_1\right)}}{\frac{1}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -0.99996)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.1)
        (* (/ t_1 (hypot t_1 (sin kx))) th)
        (if (<= t_2 0.2)
          (* (/ (sin th) (hypot (sin kx) (fabs ky))) (fabs ky))
          (if (<= t_2 0.9999)
            (/ (/ t_1 (hypot (sin kx) t_1)) (/ 1.0 th))
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * th;
	} else if (t_2 <= 0.2) {
		tmp = (sin(th) / hypot(sin(kx), fabs(ky))) * fabs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = (t_1 / hypot(sin(kx), t_1)) / (1.0 / th);
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), 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 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / Math.hypot(t_1, Math.sin(kx))) * th;
	} else if (t_2 <= 0.2) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), Math.abs(ky))) * Math.abs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = (t_1 / Math.hypot(Math.sin(kx), t_1)) / (1.0 / th);
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), 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 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_2 <= -0.99996:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.1:
		tmp = (t_1 / math.hypot(t_1, math.sin(kx))) * th
	elif t_2 <= 0.2:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), math.fabs(ky))) * math.fabs(ky)
	elif t_2 <= 0.9999:
		tmp = (t_1 / math.hypot(math.sin(kx), t_1)) / (1.0 / th)
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), 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 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.99996)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * th);
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), abs(ky))) * abs(ky));
	elseif (t_2 <= 0.9999)
		tmp = Float64(Float64(t_1 / hypot(sin(kx), t_1)) / Float64(1.0 / th));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), 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 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.99996)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.1)
		tmp = (t_1 / hypot(t_1, sin(kx))) * th;
	elseif (t_2 <= 0.2)
		tmp = (sin(th) / hypot(sin(kx), abs(ky))) * abs(ky);
	elseif (t_2 <= 0.9999)
		tmp = (t_1 / hypot(sin(kx), t_1)) / (1.0 / th);
	else
		tmp = (abs(ky) / hypot(abs(ky), 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[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -0.99996], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999], N[(N[(t$95$1 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] / N[(1.0 / th), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 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))))) < -0.99995999999999996

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 kx around 0

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

   6. Applied rewrites 57.6%

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

   if -0.99995999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.1%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 61.8%

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

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99990000000000001

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. div-flip N/A

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

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

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

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

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.7%

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 51.0%

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

      8. lift-hypot.f64 N/A

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

      9. pow2 N/A

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

      10. lift-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lower-hypot.f64 51.0%

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

   8. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow1/2 N/A

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

      14. lower-/.f64 N/A

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

   10. Applied rewrites 51.0%

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

   if 0.99990000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.6%

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

Alternative 4: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.99996:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)} \cdot \left|ky\right|\\ \mathbf{elif}\;t\_2 \leq 0.9999:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, t\_1\right)}{th \cdot t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -0.99996)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.1)
        (* (/ t_1 (hypot t_1 (sin kx))) th)
        (if (<= t_2 0.2)
          (* (/ (sin th) (hypot (sin kx) (fabs ky))) (fabs ky))
          (if (<= t_2 0.9999)
            (/ 1.0 (/ (hypot (sin kx) t_1) (* th t_1)))
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * th;
	} else if (t_2 <= 0.2) {
		tmp = (sin(th) / hypot(sin(kx), fabs(ky))) * fabs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = 1.0 / (hypot(sin(kx), t_1) / (th * t_1));
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), 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 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / Math.hypot(t_1, Math.sin(kx))) * th;
	} else if (t_2 <= 0.2) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), Math.abs(ky))) * Math.abs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = 1.0 / (Math.hypot(Math.sin(kx), t_1) / (th * t_1));
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), 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 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_2 <= -0.99996:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.1:
		tmp = (t_1 / math.hypot(t_1, math.sin(kx))) * th
	elif t_2 <= 0.2:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), math.fabs(ky))) * math.fabs(ky)
	elif t_2 <= 0.9999:
		tmp = 1.0 / (math.hypot(math.sin(kx), t_1) / (th * t_1))
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), 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 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.99996)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * th);
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), abs(ky))) * abs(ky));
	elseif (t_2 <= 0.9999)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), t_1) / Float64(th * t_1)));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), 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 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.99996)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.1)
		tmp = (t_1 / hypot(t_1, sin(kx))) * th;
	elseif (t_2 <= 0.2)
		tmp = (sin(th) / hypot(sin(kx), abs(ky))) * abs(ky);
	elseif (t_2 <= 0.9999)
		tmp = 1.0 / (hypot(sin(kx), t_1) / (th * t_1));
	else
		tmp = (abs(ky) / hypot(abs(ky), 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[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -0.99996], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision] / N[(th * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 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))))) < -0.99995999999999996

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 kx around 0

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

   6. Applied rewrites 57.6%

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

   if -0.99995999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.1%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 61.8%

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

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99990000000000001

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. div-flip N/A

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

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

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

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

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.7%

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.2%

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

   if 0.99990000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.6%

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

Alternative 5: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := \mathsf{hypot}\left(t\_1, \sin kx\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.99996:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{t\_1}{t\_3} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)} \cdot \left|ky\right|\\ \mathbf{elif}\;t\_2 \leq 0.9999:\\ \;\;\;\;\frac{th \cdot t\_1}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_3 (hypot t_1 (sin kx))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -0.99996)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.1)
        (* (/ t_1 t_3) th)
        (if (<= t_2 0.2)
          (* (/ (sin th) (hypot (sin kx) (fabs ky))) (fabs ky))
          (if (<= t_2 0.9999)
            (/ (* th t_1) t_3)
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = hypot(t_1, sin(kx));
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / t_3) * th;
	} else if (t_2 <= 0.2) {
		tmp = (sin(th) / hypot(sin(kx), fabs(ky))) * fabs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = (th * t_1) / t_3;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), 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 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_3 = Math.hypot(t_1, Math.sin(kx));
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / t_3) * th;
	} else if (t_2 <= 0.2) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), Math.abs(ky))) * Math.abs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = (th * t_1) / t_3;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), 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 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_3 = math.hypot(t_1, math.sin(kx))
	tmp = 0
	if t_2 <= -0.99996:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.1:
		tmp = (t_1 / t_3) * th
	elif t_2 <= 0.2:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), math.fabs(ky))) * math.fabs(ky)
	elif t_2 <= 0.9999:
		tmp = (th * t_1) / t_3
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), 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 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_3 = hypot(t_1, sin(kx))
	tmp = 0.0
	if (t_2 <= -0.99996)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(t_1 / t_3) * th);
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), abs(ky))) * abs(ky));
	elseif (t_2 <= 0.9999)
		tmp = Float64(Float64(th * t_1) / t_3);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), 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 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_3 = hypot(t_1, sin(kx));
	tmp = 0.0;
	if (t_2 <= -0.99996)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.1)
		tmp = (t_1 / t_3) * th;
	elseif (t_2 <= 0.2)
		tmp = (sin(th) / hypot(sin(kx), abs(ky))) * abs(ky);
	elseif (t_2 <= 0.9999)
		tmp = (th * t_1) / t_3;
	else
		tmp = (abs(ky) / hypot(abs(ky), 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[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -0.99996], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(t$95$1 / t$95$3), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999], N[(N[(th * t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 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))))) < -0.99995999999999996

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 kx around 0

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

   6. Applied rewrites 57.6%

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

   if -0.99995999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.1%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 61.8%

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

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99990000000000001

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. div-flip N/A

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

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

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

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

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.7%

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lower-/.f64 47.5%

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. lower-hypot.f64 47.5%

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

   8. Applied rewrites 47.5%

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

   if 0.99990000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.6%

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

Alternative 6: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := \mathsf{hypot}\left(t\_1, \sin kx\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.99996:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{t\_1}{t\_3} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)} \cdot \left|ky\right|\\ \mathbf{elif}\;t\_2 \leq 0.9999:\\ \;\;\;\;\frac{th \cdot t\_1}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_3 (hypot t_1 (sin kx))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -0.99996)
      (* (sin th) (copysign 1.0 t_1))
      (if (<= t_2 -0.1)
        (* (/ t_1 t_3) th)
        (if (<= t_2 0.2)
          (* (/ (sin th) (hypot (sin kx) (fabs ky))) (fabs ky))
          (if (<= t_2 0.9999)
            (/ (* th t_1) t_3)
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = hypot(t_1, sin(kx));
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / t_3) * th;
	} else if (t_2 <= 0.2) {
		tmp = (sin(th) / hypot(sin(kx), fabs(ky))) * fabs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = (th * t_1) / t_3;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), 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 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_3 = Math.hypot(t_1, Math.sin(kx));
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = Math.sin(th) * Math.copySign(1.0, t_1);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / t_3) * th;
	} else if (t_2 <= 0.2) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), Math.abs(ky))) * Math.abs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = (th * t_1) / t_3;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), 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 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_3 = math.hypot(t_1, math.sin(kx))
	tmp = 0
	if t_2 <= -0.99996:
		tmp = math.sin(th) * math.copysign(1.0, t_1)
	elif t_2 <= -0.1:
		tmp = (t_1 / t_3) * th
	elif t_2 <= 0.2:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), math.fabs(ky))) * math.fabs(ky)
	elif t_2 <= 0.9999:
		tmp = (th * t_1) / t_3
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), 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 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_3 = hypot(t_1, sin(kx))
	tmp = 0.0
	if (t_2 <= -0.99996)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(t_1 / t_3) * th);
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), abs(ky))) * abs(ky));
	elseif (t_2 <= 0.9999)
		tmp = Float64(Float64(th * t_1) / t_3);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), 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 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_3 = hypot(t_1, sin(kx));
	tmp = 0.0;
	if (t_2 <= -0.99996)
		tmp = sin(th) * (sign(t_1) * abs(1.0));
	elseif (t_2 <= -0.1)
		tmp = (t_1 / t_3) * th;
	elseif (t_2 <= 0.2)
		tmp = (sin(th) / hypot(sin(kx), abs(ky))) * abs(ky);
	elseif (t_2 <= 0.9999)
		tmp = (th * t_1) / t_3;
	else
		tmp = (abs(ky) / hypot(abs(ky), 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[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -0.99996], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(t$95$1 / t$95$3), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999], N[(N[(th * t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 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))))) < -0.99995999999999996

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

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

      10. fabs-rhs-div N/A

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

      11. lower-copysign.f64 44.3%

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

   6. Applied rewrites 44.3%

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

   if -0.99995999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.1%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 61.8%

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

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99990000000000001

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. div-flip N/A

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

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

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

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

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.7%

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lower-/.f64 47.5%

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. lower-hypot.f64 47.5%

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

   8. Applied rewrites 47.5%

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

   if 0.99990000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.6%

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

Alternative 7: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.99996:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)} \cdot \left|ky\right|\\ \mathbf{elif}\;t\_2 \leq 0.9999:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -0.99996)
      (* (sin th) (copysign 1.0 t_1))
      (if (<= t_2 -0.1)
        (* (/ t_1 (hypot t_1 (sin kx))) th)
        (if (<= t_2 0.2)
          (* (/ (sin th) (hypot (sin kx) (fabs ky))) (fabs ky))
          (if (<= t_2 0.9999)
            (* (/ th (hypot (sin kx) t_1)) t_1)
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * th;
	} else if (t_2 <= 0.2) {
		tmp = (sin(th) / hypot(sin(kx), fabs(ky))) * fabs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = (th / hypot(sin(kx), t_1)) * t_1;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), 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 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = Math.sin(th) * Math.copySign(1.0, t_1);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / Math.hypot(t_1, Math.sin(kx))) * th;
	} else if (t_2 <= 0.2) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), Math.abs(ky))) * Math.abs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = (th / Math.hypot(Math.sin(kx), t_1)) * t_1;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), 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 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_2 <= -0.99996:
		tmp = math.sin(th) * math.copysign(1.0, t_1)
	elif t_2 <= -0.1:
		tmp = (t_1 / math.hypot(t_1, math.sin(kx))) * th
	elif t_2 <= 0.2:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), math.fabs(ky))) * math.fabs(ky)
	elif t_2 <= 0.9999:
		tmp = (th / math.hypot(math.sin(kx), t_1)) * t_1
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), 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 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.99996)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * th);
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), abs(ky))) * abs(ky));
	elseif (t_2 <= 0.9999)
		tmp = Float64(Float64(th / hypot(sin(kx), t_1)) * t_1);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), 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 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.99996)
		tmp = sin(th) * (sign(t_1) * abs(1.0));
	elseif (t_2 <= -0.1)
		tmp = (t_1 / hypot(t_1, sin(kx))) * th;
	elseif (t_2 <= 0.2)
		tmp = (sin(th) / hypot(sin(kx), abs(ky))) * abs(ky);
	elseif (t_2 <= 0.9999)
		tmp = (th / hypot(sin(kx), t_1)) * t_1;
	else
		tmp = (abs(ky) / hypot(abs(ky), 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[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -0.99996], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999], N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 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))))) < -0.99995999999999996

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

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

      10. fabs-rhs-div N/A

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

      11. lower-copysign.f64 44.3%

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

   6. Applied rewrites 44.3%

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

   if -0.99995999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.1%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 61.8%

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

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99990000000000001

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

   if 0.99990000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.6%

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

Alternative 8: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := \frac{th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.99996:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)} \cdot \left|ky\right|\\ \mathbf{elif}\;t\_2 \leq 0.9999:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_3 (* (/ th (hypot (sin kx) t_1)) t_1)))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -0.99996)
      (* (sin th) (copysign 1.0 t_1))
      (if (<= t_2 -0.1)
        t_3
        (if (<= t_2 0.2)
          (* (/ (sin th) (hypot (sin kx) (fabs ky))) (fabs ky))
          (if (<= t_2 0.9999)
            t_3
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = (th / hypot(sin(kx), t_1)) * t_1;
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_2 <= -0.1) {
		tmp = t_3;
	} else if (t_2 <= 0.2) {
		tmp = (sin(th) / hypot(sin(kx), fabs(ky))) * fabs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = t_3;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), 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 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_3 = (th / Math.hypot(Math.sin(kx), t_1)) * t_1;
	double tmp;
	if (t_2 <= -0.99996) {
		tmp = Math.sin(th) * Math.copySign(1.0, t_1);
	} else if (t_2 <= -0.1) {
		tmp = t_3;
	} else if (t_2 <= 0.2) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), Math.abs(ky))) * Math.abs(ky);
	} else if (t_2 <= 0.9999) {
		tmp = t_3;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), 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 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_3 = (th / math.hypot(math.sin(kx), t_1)) * t_1
	tmp = 0
	if t_2 <= -0.99996:
		tmp = math.sin(th) * math.copysign(1.0, t_1)
	elif t_2 <= -0.1:
		tmp = t_3
	elif t_2 <= 0.2:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), math.fabs(ky))) * math.fabs(ky)
	elif t_2 <= 0.9999:
		tmp = t_3
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), 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 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_3 = Float64(Float64(th / hypot(sin(kx), t_1)) * t_1)
	tmp = 0.0
	if (t_2 <= -0.99996)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_2 <= -0.1)
		tmp = t_3;
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), abs(ky))) * abs(ky));
	elseif (t_2 <= 0.9999)
		tmp = t_3;
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), 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 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_3 = (th / hypot(sin(kx), t_1)) * t_1;
	tmp = 0.0;
	if (t_2 <= -0.99996)
		tmp = sin(th) * (sign(t_1) * abs(1.0));
	elseif (t_2 <= -0.1)
		tmp = t_3;
	elseif (t_2 <= 0.2)
		tmp = (sin(th) / hypot(sin(kx), abs(ky))) * abs(ky);
	elseif (t_2 <= 0.9999)
		tmp = t_3;
	else
		tmp = (abs(ky) / hypot(abs(ky), 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[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -0.99996], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], t$95$3, If[LessEqual[t$95$2, 0.2], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999], t$95$3, N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 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))))) < -0.99995999999999996

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

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

      10. fabs-rhs-div N/A

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

      11. lower-copysign.f64 44.3%

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

   6. Applied rewrites 44.3%

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

   if -0.99995999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99990000000000001

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 61.8%

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

   if 0.99990000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.6%

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

Alternative 9: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq -0.1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) -0.1)
      (* (sin th) (copysign 1.0 t_1))
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= -0.1) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), 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 tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= -0.1) {
		tmp = Math.sin(th) * Math.copySign(1.0, t_1);
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), 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))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= -0.1:
		tmp = math.sin(th) * math.copysign(1.0, t_1)
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= -0.1)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), 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));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= -0.1)
		tmp = sin(th) * (sign(t_1) * abs(1.0));
	else
		tmp = (abs(ky) / hypot(abs(ky), 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.1], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

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

      10. fabs-rhs-div N/A

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

      11. lower-copysign.f64 44.3%

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

   6. Applied rewrites 44.3%

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

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

   1. Initial program 94.2%

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

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.6%

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

Alternative 10: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{ky}{\frac{\left|\sin kx\right|}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (* (sin th) (copysign 1.0 (sin ky)))
     (if (<= t_1 0.05) (/ ky (/ (fabs (sin kx)) (sin th))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = sin(th) * copysign(1.0, sin(ky));
	} else if (t_1 <= 0.05) {
		tmp = ky / (fabs(sin(kx)) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = Math.sin(th) * Math.copySign(1.0, Math.sin(ky));
	} else if (t_1 <= 0.05) {
		tmp = ky / (Math.abs(Math.sin(kx)) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.1:
		tmp = math.sin(th) * math.copysign(1.0, math.sin(ky))
	elif t_1 <= 0.05:
		tmp = ky / (math.fabs(math.sin(kx)) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(sin(th) * copysign(1.0, sin(ky)));
	elseif (t_1 <= 0.05)
		tmp = Float64(ky / Float64(abs(sin(kx)) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = sin(th) * (sign(sin(ky)) * abs(1.0));
	elseif (t_1 <= 0.05)
		tmp = ky / (abs(sin(kx)) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[N[Sin[ky], $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(ky / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

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

      10. fabs-rhs-div N/A

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

      11. lower-copysign.f64 44.3%

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

   6. Applied rewrites 44.3%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 35.3%

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

   4. Applied rewrites 35.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 35.0%

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

      5. lift-sqrt.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

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

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

      9. lower-fabs.f64 37.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 37.1%

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

   6. Applied rewrites 37.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 39.1%

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

   8. Applied rewrites 39.1%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

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

Alternative 11: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\left|\sin kx\right|} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (* (sin th) (copysign 1.0 (sin ky)))
     (if (<= t_1 0.05) (* (/ (sin th) (fabs (sin kx))) ky) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = sin(th) * copysign(1.0, sin(ky));
	} else if (t_1 <= 0.05) {
		tmp = (sin(th) / fabs(sin(kx))) * ky;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = Math.sin(th) * Math.copySign(1.0, Math.sin(ky));
	} else if (t_1 <= 0.05) {
		tmp = (Math.sin(th) / Math.abs(Math.sin(kx))) * ky;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.1:
		tmp = math.sin(th) * math.copysign(1.0, math.sin(ky))
	elif t_1 <= 0.05:
		tmp = (math.sin(th) / math.fabs(math.sin(kx))) * ky
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(sin(th) * copysign(1.0, sin(ky)));
	elseif (t_1 <= 0.05)
		tmp = Float64(Float64(sin(th) / abs(sin(kx))) * ky);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = sin(th) * (sign(sin(ky)) * abs(1.0));
	elseif (t_1 <= 0.05)
		tmp = (sin(th) / abs(sin(kx))) * ky;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[N[Sin[ky], $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

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

      10. fabs-rhs-div N/A

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

      11. lower-copysign.f64 44.3%

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

   6. Applied rewrites 44.3%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 35.3%

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

   4. Applied rewrites 35.3%

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

      1. lift-/.f64 N/A

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

      2. 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 36.4%

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

      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 39.1%

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

   6. Applied rewrites 39.1%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

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

Alternative 12: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, \sin ky\right)\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (* (sin th) (copysign 1.0 (sin ky)))
     (if (<= t_1 0.05) (* (sin th) (/ ky (fabs (sin kx)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = sin(th) * copysign(1.0, sin(ky));
	} else if (t_1 <= 0.05) {
		tmp = sin(th) * (ky / fabs(sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = Math.sin(th) * Math.copySign(1.0, Math.sin(ky));
	} else if (t_1 <= 0.05) {
		tmp = Math.sin(th) * (ky / Math.abs(Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.1:
		tmp = math.sin(th) * math.copysign(1.0, math.sin(ky))
	elif t_1 <= 0.05:
		tmp = math.sin(th) * (ky / math.fabs(math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(sin(th) * copysign(1.0, sin(ky)));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(th) * Float64(ky / abs(sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = sin(th) * (sign(sin(ky)) * abs(1.0));
	elseif (t_1 <= 0.05)
		tmp = sin(th) * (ky / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[N[Sin[ky], $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

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

      10. fabs-rhs-div N/A

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

      11. lower-copysign.f64 44.3%

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

   6. Applied rewrites 44.3%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 35.3%

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

   4. Applied rewrites 35.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 36.4%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 39.1%

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

   6. Applied rewrites 39.1%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

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

Alternative 13: 60.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;\sin th \cdot \mathsf{copysign}\left(1, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sin th}{\left|kx\right|} \cdot \left|ky\right|\\ \mathbf{elif}\;t\_2 \leq 0.05:\\ \;\;\;\;\frac{\left|ky\right|}{\frac{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -0.1)
      (* (sin th) (copysign 1.0 t_1))
      (if (<= t_2 2e-120)
        (* (/ (sin th) (fabs kx)) (fabs ky))
        (if (<= t_2 0.05)
          (/ (fabs ky) (/ (hypot (fabs ky) (sin kx)) th))
          (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = sin(th) * copysign(1.0, t_1);
	} else if (t_2 <= 2e-120) {
		tmp = (sin(th) / fabs(kx)) * fabs(ky);
	} else if (t_2 <= 0.05) {
		tmp = fabs(ky) / (hypot(fabs(ky), sin(kx)) / th);
	} else {
		tmp = sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = Math.sin(th) * Math.copySign(1.0, t_1);
	} else if (t_2 <= 2e-120) {
		tmp = (Math.sin(th) / Math.abs(kx)) * Math.abs(ky);
	} else if (t_2 <= 0.05) {
		tmp = Math.abs(ky) / (Math.hypot(Math.abs(ky), Math.sin(kx)) / th);
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_2 <= -0.1:
		tmp = math.sin(th) * math.copysign(1.0, t_1)
	elif t_2 <= 2e-120:
		tmp = (math.sin(th) / math.fabs(kx)) * math.fabs(ky)
	elif t_2 <= 0.05:
		tmp = math.fabs(ky) / (math.hypot(math.fabs(ky), math.sin(kx)) / th)
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = Float64(sin(th) * copysign(1.0, t_1));
	elseif (t_2 <= 2e-120)
		tmp = Float64(Float64(sin(th) / abs(kx)) * abs(ky));
	elseif (t_2 <= 0.05)
		tmp = Float64(abs(ky) / Float64(hypot(abs(ky), sin(kx)) / th));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.1)
		tmp = sin(th) * (sign(t_1) * abs(1.0));
	elseif (t_2 <= 2e-120)
		tmp = (sin(th) / abs(kx)) * abs(ky);
	elseif (t_2 <= 0.05)
		tmp = abs(ky) / (hypot(abs(ky), sin(kx)) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -0.1], N[(N[Sin[th], $MachinePrecision] * N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t$95$1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-120], N[(N[(N[Sin[th], $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.05], N[(N[Abs[ky], $MachinePrecision] / N[(N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $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))))) < -0.10000000000000001

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

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

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

      10. fabs-rhs-div N/A

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

      11. lower-copysign.f64 44.3%

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

   6. Applied rewrites 44.3%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-120

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 35.3%

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

   4. Applied rewrites 35.3%

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

      1. lift-/.f64 N/A

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

      2. 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 36.4%

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

      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 39.1%

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

   6. Applied rewrites 39.1%

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

   7. Taylor expanded in kx around 0

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

   9. Applied rewrites 21.6%

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

   if 2e-120 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. div-flip N/A

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

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

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

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

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.7%

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 51.0%

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

      8. lift-hypot.f64 N/A

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

      9. pow2 N/A

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

      10. lift-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lower-hypot.f64 51.0%

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

   8. Applied rewrites 51.0%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 26.6%

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

   12. Taylor expanded in ky around 0

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

   14. Applied rewrites 30.5%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

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

Alternative 14: 45.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 0.05)
      (/ (fabs ky) (/ (hypot (fabs ky) (sin kx)) th))
      (sin th)))))

C

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

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 0.05) {
		tmp = Math.abs(ky) / (Math.hypot(Math.abs(ky), Math.sin(kx)) / th);
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 0.05:
		tmp = math.fabs(ky) / (math.hypot(math.fabs(ky), math.sin(kx)) / th)
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.05)
		tmp = Float64(abs(ky) / Float64(hypot(abs(ky), sin(kx)) / th));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.05)
		tmp = abs(ky) / (hypot(abs(ky), sin(kx)) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. div-flip N/A

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

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

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

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

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.7%

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

   3. Applied rewrites 95.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 51.0%

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

      8. lift-hypot.f64 N/A

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

      9. pow2 N/A

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

      10. lift-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lower-hypot.f64 51.0%

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

   8. Applied rewrites 51.0%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 26.6%

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

   12. Taylor expanded in ky around 0

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

   14. Applied rewrites 30.5%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 45.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{th}{\left|\sin kx\right|} \cdot \left|ky\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 0.05)
      (* (/ th (fabs (sin kx))) (fabs ky))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 0.05) {
		tmp = (th / fabs(sin(kx))) * fabs(ky);
	} else {
		tmp = sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 0.05) {
		tmp = (th / Math.abs(Math.sin(kx))) * Math.abs(ky);
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 0.05:
		tmp = (th / math.fabs(math.sin(kx))) * math.fabs(ky)
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.05)
		tmp = Float64(Float64(th / abs(sin(kx))) * abs(ky));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.05)
		tmp = (th / abs(sin(kx))) * abs(ky);
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 35.3%

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

   4. Applied rewrites 35.3%

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

      1. lift-/.f64 N/A

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

      2. 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 36.4%

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

      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 39.1%

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

   6. Applied rewrites 39.1%

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

   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 20.7%

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

   9. Applied rewrites 20.7%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.3%

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\left|ky\right| \cdot th}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 0.05)
      (/ (* (fabs ky) th) (fabs (sin kx)))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 0.05) {
		tmp = (fabs(ky) * th) / fabs(sin(kx));
	} else {
		tmp = sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 0.05) {
		tmp = (Math.abs(ky) * th) / Math.abs(Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 0.05:
		tmp = (math.fabs(ky) * th) / math.fabs(math.sin(kx))
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.05)
		tmp = Float64(Float64(abs(ky) * th) / abs(sin(kx)));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.05)
		tmp = (abs(ky) * th) / abs(sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.3%

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2}}}{ky \cdot \sin th}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2}}}{ky \cdot \sin th}}} \]

      4. lower-unsound-/.f64 35.0%

         \[\leadsto \frac{1}{\frac{\sqrt{{\sin kx}^{2}}}{\color{blue}{ky \cdot \sin th}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{{\sin kx}^{2}}}{\color{blue}{ky} \cdot \sin th}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{{\sin kx}^{2}}}{ky \cdot \sin th}} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx}}{ky \cdot \sin th}} \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \frac{1}{\frac{\left|\sin kx\right|}{\color{blue}{ky} \cdot \sin th}} \]

      9. lower-fabs.f64 37.1%

         \[\leadsto \frac{1}{\frac{\left|\sin kx\right|}{\color{blue}{ky} \cdot \sin th}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left|\sin kx\right|}{ky \cdot \color{blue}{\sin th}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left|\sin kx\right|}{\sin th \cdot \color{blue}{ky}}} \]

      12. lower-*.f64 37.1%

          \[\leadsto \frac{1}{\frac{\left|\sin kx\right|}{\sin th \cdot \color{blue}{ky}}} \]

   6. Applied rewrites 37.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left|\sin kx\right|}{\sin th \cdot ky}}} \]

   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 19.0%

         \[\leadsto \frac{ky \cdot th}{\left|\sin kx\right|} \]

   9. Applied rewrites 19.0%

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\left|\sin kx\right|}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 42.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin \left(\left|kx\right|\right)}^{2} + {t\_1}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\left|ky\right| \cdot \sin th}{\left|kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin (fabs kx)) 2.0) (pow t_1 2.0)))) 0.05)
      (/ (* (fabs ky) (sin th)) (fabs kx))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(fabs(kx)), 2.0) + pow(t_1, 2.0)))) <= 0.05) {
		tmp = (fabs(ky) * sin(th)) / fabs(kx);
	} else {
		tmp = sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(Math.abs(kx)), 2.0) + Math.pow(t_1, 2.0)))) <= 0.05) {
		tmp = (Math.abs(ky) * Math.sin(th)) / Math.abs(kx);
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(math.fabs(kx)), 2.0) + math.pow(t_1, 2.0)))) <= 0.05:
		tmp = (math.fabs(ky) * math.sin(th)) / math.fabs(kx)
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(abs(kx)) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.05)
		tmp = Float64(Float64(abs(ky) * sin(th)) / abs(kx));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(abs(kx)) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.05)
		tmp = (abs(ky) * sin(th)) / abs(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(N[Abs[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.3%

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{kx} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{kx} \]

      3. lower-sin.f64 15.6%

         \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   7. Applied rewrites 15.6%

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 35.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 4.2 \cdot 10^{-113}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 4.2e-113)
      (* -0.16666666666666666 (pow th 3.0))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 4.2e-113) {
		tmp = -0.16666666666666666 * pow(th, 3.0);
	} else {
		tmp = sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 4.2e-113) {
		tmp = -0.16666666666666666 * Math.pow(th, 3.0);
	} else {
		tmp = Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 4.2e-113:
		tmp = -0.16666666666666666 * math.pow(th, 3.0)
	else:
		tmp = math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 4.2e-113)
		tmp = Float64(-0.16666666666666666 * (th ^ 3.0));
	else
		tmp = sin(th);
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 4.2e-113)
		tmp = -0.16666666666666666 * (th ^ 3.0);
	else
		tmp = sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4.2e-113], N[(-0.16666666666666666 * N[Power[th, 3.0], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.2e-113

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{\color{blue}{2}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \]

      4. lower-pow.f64 13.4%

         \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

   10. Applied rewrites 13.4%

       \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]

   11. Taylor expanded in th around inf

       \[\leadsto \frac{-1}{6} \cdot {th}^{3} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot {th}^{3} \]

       2. lower-pow.f64 10.8%

          \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   13. Applied rewrites 10.8%

       \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   if 4.2e-113 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 23.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \left(\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \cdot \sin \left(\left|th\right|\right) \leq 10^{-304}:\\ \;\;\;\;-0.16666666666666666 \cdot {\left(\left|th\right|\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left|th\right| \cdot 1\\ \end{array}\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (*
     (copysign 1.0 th)
     (if (<=
          (*
           (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))
           (sin (fabs th)))
          1e-304)
       (* -0.16666666666666666 (pow (fabs th) 3.0))
       (* (fabs th) 1.0))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if (((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) * sin(fabs(th))) <= 1e-304) {
		tmp = -0.16666666666666666 * pow(fabs(th), 3.0);
	} else {
		tmp = fabs(th) * 1.0;
	}
	return copysign(1.0, ky) * (copysign(1.0, th) * tmp);
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if (((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) * Math.sin(Math.abs(th))) <= 1e-304) {
		tmp = -0.16666666666666666 * Math.pow(Math.abs(th), 3.0);
	} else {
		tmp = Math.abs(th) * 1.0;
	}
	return Math.copySign(1.0, ky) * (Math.copySign(1.0, th) * tmp);
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if ((t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) * math.sin(math.fabs(th))) <= 1e-304:
		tmp = -0.16666666666666666 * math.pow(math.fabs(th), 3.0)
	else:
		tmp = math.fabs(th) * 1.0
	return math.copysign(1.0, ky) * (math.copysign(1.0, th) * tmp)

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) * sin(abs(th))) <= 1e-304)
		tmp = Float64(-0.16666666666666666 * (abs(th) ^ 3.0));
	else
		tmp = Float64(abs(th) * 1.0);
	end
	return Float64(copysign(1.0, ky) * Float64(copysign(1.0, th) * tmp))
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if (((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) * sin(abs(th))) <= 1e-304)
		tmp = -0.16666666666666666 * (abs(th) ^ 3.0);
	else
		tmp = abs(th) * 1.0;
	end
	tmp_2 = (sign(ky) * abs(1.0)) * ((sign(th) * abs(1.0)) * tmp);
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-304], N[(-0.16666666666666666 * N[Power[N[Abs[th], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[Abs[th], $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 9.9999999999999997e-305

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{\color{blue}{2}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \]

      4. lower-pow.f64 13.4%

         \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

   10. Applied rewrites 13.4%

       \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]

   11. Taylor expanded in th around inf

       \[\leadsto \frac{-1}{6} \cdot {th}^{3} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot {th}^{3} \]

       2. lower-pow.f64 10.8%

          \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   13. Applied rewrites 10.8%

       \[\leadsto -0.16666666666666666 \cdot {th}^{3} \]

   if 9.9999999999999997e-305 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

      7. lower-sin.f64 41.3%

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   4. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \sin th \]
   6. Step-by-step derivation

      1. lower-sin.f64 23.7%

         \[\leadsto \sin th \]

   7. Applied rewrites 23.7%

      \[\leadsto \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{\color{blue}{2}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \]

      4. lower-pow.f64 13.4%

         \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

   10. Applied rewrites 13.4%

       \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]

   11. Taylor expanded in th around 0

       \[\leadsto th \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 13.7%

       \[\leadsto th \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 16.1% accurate, 18.8× speedup

Math

\[\mathsf{copysign}\left(1, ky\right) \cdot \left(th \cdot 1\right) \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (copysign 1.0 ky) (* th 1.0)))

C

double code(double kx, double ky, double th) {
	return copysign(1.0, ky) * (th * 1.0);
}

Java

public static double code(double kx, double ky, double th) {
	return Math.copySign(1.0, ky) * (th * 1.0);
}

Python

def code(kx, ky, th):
	return math.copysign(1.0, ky) * (th * 1.0)

Julia

function code(kx, ky, th)
	return Float64(copysign(1.0, ky) * Float64(th * 1.0))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sign(ky) * abs(1.0)) * (th * 1.0);
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(th * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\color{blue}{\sin ky}}^{2}}} \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{\color{blue}{2}}}} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

   7. lower-sin.f64 41.3%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}} \]

4. Applied rewrites 41.3%

   \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}} \]

5. Taylor expanded in ky around 0

   \[\leadsto \sin th \]
6. Step-by-step derivation

   1. lower-sin.f64 23.7%

      \[\leadsto \sin th \]

7. Applied rewrites 23.7%

   \[\leadsto \sin th \]

8. Taylor expanded in th around 0

   \[\leadsto th \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {th}^{2}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{th}^{2}}\right) \]

   2. lower-+.f64 N/A

      \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{\color{blue}{2}}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \]

   4. lower-pow.f64 13.4%

      \[\leadsto th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right) \]

10. Applied rewrites 13.4%

    \[\leadsto th \cdot \left(1 + \color{blue}{-0.16666666666666666 \cdot {th}^{2}}\right) \]

11. Taylor expanded in th around 0

    \[\leadsto th \cdot 1 \]
12. Step-by-step derivation

13. Applied rewrites 13.7%

    \[\leadsto th \cdot 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025205 
(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)))