Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 7.1s

Alternatives: 17

Speedup: 1.2×

Specification

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Initial Program: 93.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (/ (sin th) (/ (hypot (sin ky) (sin kx)) (sin ky))))

C

double code(double kx, double ky, double th) {
	return sin(th) / (hypot(sin(ky), sin(kx)) / sin(ky));
}

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th) / (Math.hypot(Math.sin(ky), Math.sin(kx)) / Math.sin(ky));
}

Python

def code(kx, ky, th):
	return math.sin(th) / (math.hypot(math.sin(ky), math.sin(kx)) / math.sin(ky))

Julia

function code(kx, ky, th)
	return Float64(sin(th) / Float64(hypot(sin(ky), sin(kx)) / sin(ky)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th) / (hypot(sin(ky), sin(kx)) / sin(ky));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f64 93.7

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow2 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-hypot.f64 99.6

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

3. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.7

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

   6. lift-hypot.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-hypot.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (hypot (sin ky) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

3. Applied rewrites 99.7%

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

Alternative 3: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{t\_1}{\sqrt{t\_2 + {t\_1}^{2}}}\\ t_4 := t\_3 \cdot th\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 10^{-5}:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_2 + {\left(\left|ky\right|\right)}^{2}}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.998:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)}{\left|ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ t_1 (sqrt (+ t_2 (pow t_1 2.0)))))
        (t_4 (* t_3 th)))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -1.0)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_3 -0.2)
        t_4
        (if (<= t_3 1e-5)
          (* (/ t_1 (sqrt (+ t_2 (pow (fabs ky) 2.0)))) (sin th))
          (if (<= t_3 0.998)
            t_4
            (/ (sin th) (/ (hypot (fabs ky) (sin kx)) (fabs ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = t_1 / sqrt((t_2 + pow(t_1, 2.0)));
	double t_4 = t_3 * th;
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 1e-5) {
		tmp = (t_1 / sqrt((t_2 + pow(fabs(ky), 2.0)))) * sin(th);
	} else if (t_3 <= 0.998) {
		tmp = t_4;
	} else {
		tmp = sin(th) / (hypot(fabs(ky), sin(kx)) / fabs(ky));
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = t_1 / Math.sqrt((t_2 + Math.pow(t_1, 2.0)));
	double t_4 = t_3 * th;
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 1e-5) {
		tmp = (t_1 / Math.sqrt((t_2 + Math.pow(Math.abs(ky), 2.0)))) * Math.sin(th);
	} else if (t_3 <= 0.998) {
		tmp = t_4;
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.abs(ky), Math.sin(kx)) / Math.abs(ky));
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = t_1 / math.sqrt((t_2 + math.pow(t_1, 2.0)))
	t_4 = t_3 * th
	tmp = 0
	if t_3 <= -1.0:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_3 <= -0.2:
		tmp = t_4
	elif t_3 <= 1e-5:
		tmp = (t_1 / math.sqrt((t_2 + math.pow(math.fabs(ky), 2.0)))) * math.sin(th)
	elif t_3 <= 0.998:
		tmp = t_4
	else:
		tmp = math.sin(th) / (math.hypot(math.fabs(ky), math.sin(kx)) / math.fabs(ky))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(t_1 / sqrt(Float64(t_2 + (t_1 ^ 2.0))))
	t_4 = Float64(t_3 * th)
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 1e-5)
		tmp = Float64(Float64(t_1 / sqrt(Float64(t_2 + (abs(ky) ^ 2.0)))) * sin(th));
	elseif (t_3 <= 0.998)
		tmp = t_4;
	else
		tmp = Float64(sin(th) / Float64(hypot(abs(ky), sin(kx)) / abs(ky)));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = sin(kx) ^ 2.0;
	t_3 = t_1 / sqrt((t_2 + (t_1 ^ 2.0)));
	t_4 = t_3 * th;
	tmp = 0.0;
	if (t_3 <= -1.0)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 1e-5)
		tmp = (t_1 / sqrt((t_2 + (abs(ky) ^ 2.0)))) * sin(th);
	elseif (t_3 <= 0.998)
		tmp = t_4;
	else
		tmp = sin(th) / (hypot(abs(ky), sin(kx)) / abs(ky));
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(t$95$2 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * th), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -1.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], t$95$4, If[LessEqual[t$95$3, 1e-5], N[(N[(t$95$1 / N[Sqrt[N[(t$95$2 + N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.998], t$95$4, N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in 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 58.2%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 93.7%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 47.8%

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

   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))))) < 1.00000000000000008e-5

   1. Initial program 93.7%

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

   2. Taylor expanded in ky around 0

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

      1. lower-pow.f64 47.1

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

   4. Applied rewrites 47.1%

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

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

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 93.7

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.7

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-hypot.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 65.8%

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

Alternative 4: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := t\_2 \cdot th\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;t\_2 \leq 0.998:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)}{\left|ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_3 (* t_2 th)))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -1.0)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.2)
        t_3
        (if (<= t_2 1e-5)
          (* t_1 (/ (sin th) (fabs (sin kx))))
          (if (<= t_2 0.998)
            t_3
            (/ (sin th) (/ (hypot (fabs ky) (sin kx)) (fabs ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = t_2 * th;
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 1e-5) {
		tmp = t_1 * (sin(th) / fabs(sin(kx)));
	} else if (t_2 <= 0.998) {
		tmp = t_3;
	} else {
		tmp = sin(th) / (hypot(fabs(ky), sin(kx)) / fabs(ky));
	}
	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 = t_2 * th;
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 1e-5) {
		tmp = t_1 * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else if (t_2 <= 0.998) {
		tmp = t_3;
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.abs(ky), Math.sin(kx)) / Math.abs(ky));
	}
	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 = t_2 * th
	tmp = 0
	if t_2 <= -1.0:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.2:
		tmp = t_3
	elif t_2 <= 1e-5:
		tmp = t_1 * (math.sin(th) / math.fabs(math.sin(kx)))
	elif t_2 <= 0.998:
		tmp = t_3
	else:
		tmp = math.sin(th) / (math.hypot(math.fabs(ky), math.sin(kx)) / math.fabs(ky))
	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(t_2 * th)
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 1e-5)
		tmp = Float64(t_1 * Float64(sin(th) / abs(sin(kx))));
	elseif (t_2 <= 0.998)
		tmp = t_3;
	else
		tmp = Float64(sin(th) / Float64(hypot(abs(ky), sin(kx)) / abs(ky)));
	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 = t_2 * th;
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 1e-5)
		tmp = t_1 * (sin(th) / abs(sin(kx)));
	elseif (t_2 <= 0.998)
		tmp = t_3;
	else
		tmp = sin(th) / (hypot(abs(ky), sin(kx)) / abs(ky));
	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[(t$95$2 * th), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], t$95$3, If[LessEqual[t$95$2, 1e-5], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.998], t$95$3, N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in 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 58.2%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 93.7%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 47.8%

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

   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))))) < 1.00000000000000008e-5

   1. Initial program 93.7%

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

   2. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.9

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

   4. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 45.0

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

   6. Applied rewrites 45.0%

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

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

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 93.7

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.7

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-hypot.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 65.8%

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

Alternative 5: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := \mathsf{hypot}\left(t\_1, \sin kx\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;\frac{t\_1}{t\_3} \cdot th\\ \mathbf{elif}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;t\_2 \leq 0.998:\\ \;\;\;\;\frac{t\_1 \cdot th}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)}{\left|ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_3 (hypot t_1 (sin kx))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -1.0)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.2)
        (* (/ t_1 t_3) th)
        (if (<= t_2 1e-5)
          (* t_1 (/ (sin th) (fabs (sin kx))))
          (if (<= t_2 0.998)
            (/ (* t_1 th) t_3)
            (/ (sin th) (/ (hypot (fabs ky) (sin kx)) (fabs ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = hypot(t_1, sin(kx));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.2) {
		tmp = (t_1 / t_3) * th;
	} else if (t_2 <= 1e-5) {
		tmp = t_1 * (sin(th) / fabs(sin(kx)));
	} else if (t_2 <= 0.998) {
		tmp = (t_1 * th) / t_3;
	} else {
		tmp = sin(th) / (hypot(fabs(ky), sin(kx)) / fabs(ky));
	}
	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 <= -1.0) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.2) {
		tmp = (t_1 / t_3) * th;
	} else if (t_2 <= 1e-5) {
		tmp = t_1 * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else if (t_2 <= 0.998) {
		tmp = (t_1 * th) / t_3;
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.abs(ky), Math.sin(kx)) / Math.abs(ky));
	}
	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 <= -1.0:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.2:
		tmp = (t_1 / t_3) * th
	elif t_2 <= 1e-5:
		tmp = t_1 * (math.sin(th) / math.fabs(math.sin(kx)))
	elif t_2 <= 0.998:
		tmp = (t_1 * th) / t_3
	else:
		tmp = math.sin(th) / (math.hypot(math.fabs(ky), math.sin(kx)) / math.fabs(ky))
	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 <= -1.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.2)
		tmp = Float64(Float64(t_1 / t_3) * th);
	elseif (t_2 <= 1e-5)
		tmp = Float64(t_1 * Float64(sin(th) / abs(sin(kx))));
	elseif (t_2 <= 0.998)
		tmp = Float64(Float64(t_1 * th) / t_3);
	else
		tmp = Float64(sin(th) / Float64(hypot(abs(ky), sin(kx)) / abs(ky)));
	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 <= -1.0)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.2)
		tmp = (t_1 / t_3) * th;
	elseif (t_2 <= 1e-5)
		tmp = t_1 * (sin(th) / abs(sin(kx)));
	elseif (t_2 <= 0.998)
		tmp = (t_1 * th) / t_3;
	else
		tmp = sin(th) / (hypot(abs(ky), sin(kx)) / abs(ky));
	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, -1.0], 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, -0.2], N[(N[(t$95$1 / t$95$3), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 1e-5], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.998], N[(N[(t$95$1 * th), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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 58.2%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.7%

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

   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))))) < 1.00000000000000008e-5

   1. Initial program 93.7%

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

   2. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.9

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

   4. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 45.0

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

   6. Applied rewrites 45.0%

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

   if 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 93.7%

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

      1. lift-*.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. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 47.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 47.1

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

      9. lift-hypot.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-hypot.f64 47.1

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

   8. Applied rewrites 47.1%

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

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

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 93.7

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.7

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-hypot.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 65.8%

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

Alternative 6: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := \frac{t\_1 \cdot th}{\mathsf{hypot}\left(t\_1, \sin kx\right)}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;t\_2 \leq 0.998:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)}{\left|ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_3 (/ (* t_1 th) (hypot t_1 (sin kx)))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -1.0)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.2)
        t_3
        (if (<= t_2 1e-5)
          (* t_1 (/ (sin th) (fabs (sin kx))))
          (if (<= t_2 0.998)
            t_3
            (/ (sin th) (/ (hypot (fabs ky) (sin kx)) (fabs ky))))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = (t_1 * th) / hypot(t_1, sin(kx));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 1e-5) {
		tmp = t_1 * (sin(th) / fabs(sin(kx)));
	} else if (t_2 <= 0.998) {
		tmp = t_3;
	} else {
		tmp = sin(th) / (hypot(fabs(ky), sin(kx)) / fabs(ky));
	}
	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 = (t_1 * th) / Math.hypot(t_1, Math.sin(kx));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 1e-5) {
		tmp = t_1 * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else if (t_2 <= 0.998) {
		tmp = t_3;
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.abs(ky), Math.sin(kx)) / Math.abs(ky));
	}
	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 = (t_1 * th) / math.hypot(t_1, math.sin(kx))
	tmp = 0
	if t_2 <= -1.0:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.2:
		tmp = t_3
	elif t_2 <= 1e-5:
		tmp = t_1 * (math.sin(th) / math.fabs(math.sin(kx)))
	elif t_2 <= 0.998:
		tmp = t_3
	else:
		tmp = math.sin(th) / (math.hypot(math.fabs(ky), math.sin(kx)) / math.fabs(ky))
	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(t_1 * th) / hypot(t_1, sin(kx)))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 1e-5)
		tmp = Float64(t_1 * Float64(sin(th) / abs(sin(kx))));
	elseif (t_2 <= 0.998)
		tmp = t_3;
	else
		tmp = Float64(sin(th) / Float64(hypot(abs(ky), sin(kx)) / abs(ky)));
	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 = (t_1 * th) / hypot(t_1, sin(kx));
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 1e-5)
		tmp = t_1 * (sin(th) / abs(sin(kx)));
	elseif (t_2 <= 0.998)
		tmp = t_3;
	else
		tmp = sin(th) / (hypot(abs(ky), sin(kx)) / abs(ky));
	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[(t$95$1 * th), $MachinePrecision] / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], t$95$3, If[LessEqual[t$95$2, 1e-5], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.998], t$95$3, N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in 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 58.2%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 93.7%

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

      1. lift-*.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. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 47.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 47.1

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

      9. lift-hypot.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-hypot.f64 47.1

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

   8. Applied rewrites 47.1%

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

   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))))) < 1.00000000000000008e-5

   1. Initial program 93.7%

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

   2. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.9

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

   4. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 45.0

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

   6. Applied rewrites 45.0%

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

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

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 93.7

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.7

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-hypot.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 65.8%

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

Alternative 7: 81.8% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 7.5 \cdot 10^{-13}:\\ \;\;\;\;\sin ky \cdot \frac{\left|th\right|}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\left|th\right|\right)}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (copysign 1.0 th)
  (if (<= (fabs th) 7.5e-13)
    (* (sin ky) (/ (fabs th) (hypot (sin kx) (sin ky))))
    (/ (sin (fabs th)) (/ (hypot ky (sin kx)) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 7.5e-13) {
		tmp = sin(ky) * (fabs(th) / hypot(sin(kx), sin(ky)));
	} else {
		tmp = sin(fabs(th)) / (hypot(ky, sin(kx)) / ky);
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 7.5e-13) {
		tmp = Math.sin(ky) * (Math.abs(th) / Math.hypot(Math.sin(kx), Math.sin(ky)));
	} else {
		tmp = Math.sin(Math.abs(th)) / (Math.hypot(ky, Math.sin(kx)) / ky);
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 7.5e-13:
		tmp = math.sin(ky) * (math.fabs(th) / math.hypot(math.sin(kx), math.sin(ky)))
	else:
		tmp = math.sin(math.fabs(th)) / (math.hypot(ky, math.sin(kx)) / ky)
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 7.5e-13)
		tmp = Float64(sin(ky) * Float64(abs(th) / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(sin(abs(th)) / Float64(hypot(ky, sin(kx)) / ky));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 7.5e-13)
		tmp = sin(ky) * (abs(th) / hypot(sin(kx), sin(ky)));
	else
		tmp = sin(abs(th)) / (hypot(ky, sin(kx)) / ky);
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 7.5e-13], N[(N[Sin[ky], $MachinePrecision] * N[(N[Abs[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 7.5000000000000004e-13

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 50.7

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

      7. lift-hypot.f64 N/A

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

      8. pow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lift-sin.f64 N/A

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

      13. sqrt-fabs-rev N/A

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

      14. lift-sin.f64 N/A

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

      15. pow2 N/A

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

   8. Applied rewrites 50.7%

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

   if 7.5000000000000004e-13 < th

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 93.7

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.7

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-hypot.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 65.8%

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

Alternative 8: 80.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1 \cdot th}{\mathsf{hypot}\left(t\_1, \sin kx\right)}\\ t_3 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-5}:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;t\_3 \leq 0.998:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)}{\left|ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ (* t_1 th) (hypot t_1 (sin kx))))
        (t_3 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.2)
      t_2
      (if (<= t_3 1e-5)
        (* t_1 (/ (sin th) (fabs (sin kx))))
        (if (<= t_3 0.998)
          t_2
          (/ (sin th) (/ (hypot (fabs ky) (sin kx)) (fabs ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = (t_1 * th) / hypot(t_1, sin(kx));
	double t_3 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= -0.2) {
		tmp = t_2;
	} else if (t_3 <= 1e-5) {
		tmp = t_1 * (sin(th) / fabs(sin(kx)));
	} else if (t_3 <= 0.998) {
		tmp = t_2;
	} else {
		tmp = sin(th) / (hypot(fabs(ky), sin(kx)) / fabs(ky));
	}
	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 * th) / Math.hypot(t_1, Math.sin(kx));
	double t_3 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= -0.2) {
		tmp = t_2;
	} else if (t_3 <= 1e-5) {
		tmp = t_1 * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else if (t_3 <= 0.998) {
		tmp = t_2;
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.abs(ky), Math.sin(kx)) / Math.abs(ky));
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = (t_1 * th) / math.hypot(t_1, math.sin(kx))
	t_3 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_3 <= -0.2:
		tmp = t_2
	elif t_3 <= 1e-5:
		tmp = t_1 * (math.sin(th) / math.fabs(math.sin(kx)))
	elif t_3 <= 0.998:
		tmp = t_2
	else:
		tmp = math.sin(th) / (math.hypot(math.fabs(ky), math.sin(kx)) / math.fabs(ky))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(t_1 * th) / hypot(t_1, sin(kx)))
	t_3 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.2)
		tmp = t_2;
	elseif (t_3 <= 1e-5)
		tmp = Float64(t_1 * Float64(sin(th) / abs(sin(kx))));
	elseif (t_3 <= 0.998)
		tmp = t_2;
	else
		tmp = Float64(sin(th) / Float64(hypot(abs(ky), sin(kx)) / abs(ky)));
	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 * th) / hypot(t_1, sin(kx));
	t_3 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -0.2)
		tmp = t_2;
	elseif (t_3 <= 1e-5)
		tmp = t_1 * (sin(th) / abs(sin(kx)));
	elseif (t_3 <= 0.998)
		tmp = t_2;
	else
		tmp = sin(th) / (hypot(abs(ky), sin(kx)) / abs(ky));
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * th), $MachinePrecision] / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.2], t$95$2, If[LessEqual[t$95$3, 1e-5], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.998], t$95$2, N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 93.7%

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

      1. lift-*.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. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 47.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 47.1

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

      9. lift-hypot.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-hypot.f64 47.1

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

   8. Applied rewrites 47.1%

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

   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))))) < 1.00000000000000008e-5

   1. Initial program 93.7%

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

   2. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.9

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

   4. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 45.0

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

   6. Applied rewrites 45.0%

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

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

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 93.7

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.7

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-hypot.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 65.8%

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

Alternative 9: 72.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = pow(t_1, 2.0);
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + t_2))) <= -0.2) {
		tmp = (t_1 / sqrt(t_2)) * th;
	} else {
		tmp = sin(th) / (hypot(fabs(ky), sin(kx)) / fabs(ky));
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = Math.pow(t_1, 2.0);
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_2))) <= -0.2) {
		tmp = (t_1 / Math.sqrt(t_2)) * th;
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.abs(ky), Math.sin(kx)) / Math.abs(ky));
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.pow(t_1, 2.0)
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + t_2))) <= -0.2:
		tmp = (t_1 / math.sqrt(t_2)) * th
	else:
		tmp = math.sin(th) / (math.hypot(math.fabs(ky), math.sin(kx)) / math.fabs(ky))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = t_1 ^ 2.0
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + t_2))) <= -0.2)
		tmp = Float64(Float64(t_1 / sqrt(t_2)) * th);
	else
		tmp = Float64(sin(th) / Float64(hypot(abs(ky), sin(kx)) / abs(ky)));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 ^ 2.0;
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + t_2))) <= -0.2)
		tmp = (t_1 / sqrt(t_2)) * th;
	else
		tmp = sin(th) / (hypot(abs(ky), sin(kx)) / abs(ky));
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, 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] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(t$95$1 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $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))))) < -0.20000000000000001

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.7%

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

   7. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 21.0

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

   9. Applied rewrites 21.0%

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

   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)))))

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 93.7

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.7

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-hypot.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 65.8%

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

Alternative 10: 71.8% accurate, 0.7× speedup

Math

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

FPCore

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

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)))) <= -0.2) {
		tmp = (1.0 / hypot(kx, t_1)) * (th * t_1);
	} else {
		tmp = sin(th) / (hypot(fabs(ky), sin(kx)) / fabs(ky));
	}
	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)))) <= -0.2) {
		tmp = (1.0 / Math.hypot(kx, t_1)) * (th * t_1);
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.abs(ky), Math.sin(kx)) / Math.abs(ky));
	}
	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)))) <= -0.2:
		tmp = (1.0 / math.hypot(kx, t_1)) * (th * t_1)
	else:
		tmp = math.sin(th) / (math.hypot(math.fabs(ky), math.sin(kx)) / math.fabs(ky))
	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)))) <= -0.2)
		tmp = Float64(Float64(1.0 / hypot(kx, t_1)) * Float64(th * t_1));
	else
		tmp = Float64(sin(th) / Float64(hypot(abs(ky), sin(kx)) / abs(ky)));
	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)))) <= -0.2)
		tmp = (1.0 / hypot(kx, t_1)) * (th * t_1);
	else
		tmp = sin(th) / (hypot(abs(ky), sin(kx)) / abs(ky));
	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], -0.2], N[(N[(1.0 / N[Sqrt[kx ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $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))))) < -0.20000000000000001

   1. Initial program 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.1%

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

   7. Taylor expanded in kx around 0

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

   9. Applied rewrites 30.1%

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

   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)))))

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 93.7

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.7

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-hypot.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 65.8%

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

Alternative 11: 67.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_1 -0.05)
      (* (* th t_1) (/ 1.0 (fabs (sin kx))))
      (/ (sin th) (/ (hypot (fabs ky) (sin kx)) (fabs ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (th * t_1) * (1.0 / fabs(sin(kx)));
	} else {
		tmp = sin(th) / (hypot(fabs(ky), sin(kx)) / fabs(ky));
	}
	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 <= -0.05) {
		tmp = (th * t_1) * (1.0 / Math.abs(Math.sin(kx)));
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.abs(ky), Math.sin(kx)) / Math.abs(ky));
	}
	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 <= -0.05:
		tmp = (th * t_1) * (1.0 / math.fabs(math.sin(kx)))
	else:
		tmp = math.sin(th) / (math.hypot(math.fabs(ky), math.sin(kx)) / math.fabs(ky))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(Float64(th * t_1) * Float64(1.0 / abs(sin(kx))));
	else
		tmp = Float64(sin(th) / Float64(hypot(abs(ky), sin(kx)) / abs(ky)));
	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 <= -0.05)
		tmp = (th * t_1) * (1.0 / abs(sin(kx)));
	else
		tmp = sin(th) / (hypot(abs(ky), sin(kx)) / abs(ky));
	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[t$95$1, -0.05], N[(N[(th * t$95$1), $MachinePrecision] * N[(1.0 / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.050000000000000003

   1. Initial program 93.7%

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

   2. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.9

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

   4. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. mult-flip N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 40.9

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

      9. lift-sqrt.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

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

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

   6. Applied rewrites 43.2%

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

   7. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 22.1

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

   9. Applied rewrites 22.1%

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

   if -0.050000000000000003 < (sin.f64 ky)

   1. Initial program 93.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 93.7

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.7

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-hypot.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 65.8%

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

Alternative 12: 67.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_1 -0.05)
      (* (* th t_1) (/ 1.0 (fabs (sin kx))))
      (* (/ (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 <= -0.05) {
		tmp = (th * t_1) * (1.0 / fabs(sin(kx)));
	} 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 <= -0.05) {
		tmp = (th * t_1) * (1.0 / Math.abs(Math.sin(kx)));
	} 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 <= -0.05:
		tmp = (th * t_1) * (1.0 / math.fabs(math.sin(kx)))
	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 (t_1 <= -0.05)
		tmp = Float64(Float64(th * t_1) * Float64(1.0 / abs(sin(kx))));
	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 <= -0.05)
		tmp = (th * t_1) * (1.0 / abs(sin(kx)));
	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[t$95$1, -0.05], N[(N[(th * t$95$1), $MachinePrecision] * N[(1.0 / N[Abs[N[Sin[kx], $MachinePrecision]], $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 2 regimes

2. if (sin.f64 ky) < -0.050000000000000003

   1. Initial program 93.7%

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

   2. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.9

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

   4. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. mult-flip N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 40.9

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

      9. lift-sqrt.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

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

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

   6. Applied rewrites 43.2%

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

   7. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 22.1

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

   9. Applied rewrites 22.1%

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

   if -0.050000000000000003 < (sin.f64 ky)

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.9%

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

   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 65.8%

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

Alternative 13: 52.5% accurate, 2.0× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \left|th\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left|\sin kx\right|}{ky}} \cdot \sin \left(\left|th\right|\right)\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (copysign 1.0 th)
  (if (<= (fabs th) 6e-9)
    (* (/ ky (hypot ky (sin kx))) (fabs th))
    (* (/ 1.0 (/ (fabs (sin kx)) ky)) (sin (fabs th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 6e-9) {
		tmp = (ky / hypot(ky, sin(kx))) * fabs(th);
	} else {
		tmp = (1.0 / (fabs(sin(kx)) / ky)) * sin(fabs(th));
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 6e-9) {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.abs(th);
	} else {
		tmp = (1.0 / (Math.abs(Math.sin(kx)) / ky)) * Math.sin(Math.abs(th));
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 6e-9:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.fabs(th)
	else:
		tmp = (1.0 / (math.fabs(math.sin(kx)) / ky)) * math.sin(math.fabs(th))
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 6e-9)
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * abs(th));
	else
		tmp = Float64(Float64(1.0 / Float64(abs(sin(kx)) / ky)) * sin(abs(th)));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 6e-9)
		tmp = (ky / hypot(ky, sin(kx))) * abs(th);
	else
		tmp = (1.0 / (abs(sin(kx)) / ky)) * sin(abs(th));
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 6e-9], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Abs[th], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 5.99999999999999996e-9

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 27.2%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 34.3%

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

   if 5.99999999999999996e-9 < th

   1. Initial program 93.7%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.9

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

   4. Applied rewrites 36.9%

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

      1. lift-/.f64 N/A

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

      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 36.9

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

      5. lift-sqrt.f64 N/A

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

      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

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

      9. lower-fabs.f64 39.9

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

   6. Applied rewrites 39.9%

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

Alternative 14: 52.5% accurate, 2.0× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \left|th\right|\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|th\right|\right) \cdot \frac{ky}{\left|\sin kx\right|}\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (copysign 1.0 th)
  (if (<= (fabs th) 6e-9)
    (* (/ ky (hypot ky (sin kx))) (fabs th))
    (* (sin (fabs th)) (/ ky (fabs (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 6e-9) {
		tmp = (ky / hypot(ky, sin(kx))) * fabs(th);
	} else {
		tmp = sin(fabs(th)) * (ky / fabs(sin(kx)));
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 6e-9) {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.abs(th);
	} else {
		tmp = Math.sin(Math.abs(th)) * (ky / Math.abs(Math.sin(kx)));
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 6e-9:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.fabs(th)
	else:
		tmp = math.sin(math.fabs(th)) * (ky / math.fabs(math.sin(kx)))
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 6e-9)
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * abs(th));
	else
		tmp = Float64(sin(abs(th)) * Float64(ky / abs(sin(kx))));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 6e-9)
		tmp = (ky / hypot(ky, sin(kx))) * abs(th);
	else
		tmp = sin(abs(th)) * (ky / abs(sin(kx)));
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 6e-9], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Abs[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision] * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 5.99999999999999996e-9

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 27.2%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 34.3%

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

   if 5.99999999999999996e-9 < th

   1. Initial program 93.7%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.9

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

   4. Applied rewrites 36.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.9

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

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

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

      8. lower-fabs.f64 39.9

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

   6. Applied rewrites 39.9%

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

Alternative 15: 40.7% accurate, 2.6× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 620:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin \left(\left|kx\right|\right)\right)} \cdot \left|th\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left|kx\right|}{ky}} \cdot \sin \left(\left|th\right|\right)\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (copysign 1.0 th)
  (if (<= (fabs th) 620.0)
    (* (/ ky (hypot ky (sin (fabs kx)))) (fabs th))
    (* (/ 1.0 (/ (fabs kx) ky)) (sin (fabs th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 620.0) {
		tmp = (ky / hypot(ky, sin(fabs(kx)))) * fabs(th);
	} else {
		tmp = (1.0 / (fabs(kx) / ky)) * sin(fabs(th));
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 620.0) {
		tmp = (ky / Math.hypot(ky, Math.sin(Math.abs(kx)))) * Math.abs(th);
	} else {
		tmp = (1.0 / (Math.abs(kx) / ky)) * Math.sin(Math.abs(th));
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 620.0:
		tmp = (ky / math.hypot(ky, math.sin(math.fabs(kx)))) * math.fabs(th)
	else:
		tmp = (1.0 / (math.fabs(kx) / ky)) * math.sin(math.fabs(th))
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 620.0)
		tmp = Float64(Float64(ky / hypot(ky, sin(abs(kx)))) * abs(th));
	else
		tmp = Float64(Float64(1.0 / Float64(abs(kx) / ky)) * sin(abs(th)));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 620.0)
		tmp = (ky / hypot(ky, sin(abs(kx)))) * abs(th);
	else
		tmp = (1.0 / (abs(kx) / ky)) * sin(abs(th));
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 620.0], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Abs[th], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Abs[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 620

   1. Initial program 93.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 27.2%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 34.3%

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

   if 620 < th

   1. Initial program 93.7%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.9

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

   4. Applied rewrites 36.9%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 17.0

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

   7. Applied rewrites 17.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 17.0

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

   9. Applied rewrites 17.0%

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

Alternative 16: 22.5% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ ky (fabs kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	return (ky / fabs(kx)) * 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 = (ky / abs(kx)) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (ky / Math.abs(kx)) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (ky / math.fabs(kx)) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(ky / abs(kx)) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (ky / abs(kx)) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

2. Taylor expanded in ky around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 36.9

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

4. Applied rewrites 36.9%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 17.0

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

7. Applied rewrites 17.0%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
8. Add Preprocessing

Alternative 17: 16.0% accurate, 14.9× speedup

Math

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

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ ky (fabs kx)) (* th 1.0)))

C

double code(double kx, double ky, double th) {
	return (ky / fabs(kx)) * (th * 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (ky / abs(kx)) * (th * 1.0d0)
end function

Java

public static double code(double kx, double ky, double th) {
	return (ky / Math.abs(kx)) * (th * 1.0);
}

Python

def code(kx, ky, th):
	return (ky / math.fabs(kx)) * (th * 1.0)

Julia

function code(kx, ky, th)
	return Float64(Float64(ky / abs(kx)) * Float64(th * 1.0))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (ky / abs(kx)) * (th * 1.0);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[(th * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

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

2. Taylor expanded in ky around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 36.9

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

4. Applied rewrites 36.9%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 17.0

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

7. Applied rewrites 17.0%

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

8. Taylor expanded in th around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 13.2

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

10. Applied rewrites 13.2%

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

11. Taylor expanded in th around 0

    \[\leadsto \frac{ky}{kx} \cdot \left(th \cdot 1\right) \]
12. Step-by-step derivation

13. Applied rewrites 13.8%

    \[\leadsto \frac{ky}{kx} \cdot \left(th \cdot 1\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025168 
(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)))