Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.6% → 99.7%

Time: 7.8s

Alternatives: 23

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. lift-sqrt.64 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.64 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.64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7%

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

3. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-hypot.f64 N/A

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

   7. pow2 N/A

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

   8. lift-pow.64 N/A

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

   9. +-commutative N/A

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

   10. lift-pow.64 N/A

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

   11. pow2 N/A

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

   12. lift-hypot.f64 N/A

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

   13. div-flip-rev N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-/r* N/A

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

   17. div-flip-rev N/A

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

5. Applied rewrites 99.6%

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

Alternative 2: 99.6% 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 93.6%

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

   1. lift-sqrt.64 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.64 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.64 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: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)}\\ t_3 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_4 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(kx \cdot \left(1 + -0.16666666666666666 \cdot {kx}^{2}\right), t\_1\right)} \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_2 \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_4}{\mathsf{hypot}\left(t\_4, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;t\_2 \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (hypot t_1 (sin kx))))
        (t_3 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_4
         (* (fabs ky) (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.98)
      (*
       (/
        (sin th)
        (hypot (* kx (+ 1.0 (* -0.16666666666666666 (pow kx 2.0)))) t_1))
       t_1)
      (if (<= t_3 -0.2)
        (* t_2 (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0)))))
        (if (<= t_3 2e-7)
          (* (/ t_4 (hypot t_4 (sin kx))) (sin th))
          (if (<= t_3 0.995)
            (* t_2 th)
            (* (/ t_1 (hypot t_1 kx)) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / hypot(t_1, sin(kx));
	double t_3 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_4 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (sin(th) / hypot((kx * (1.0 + (-0.16666666666666666 * pow(kx, 2.0)))), t_1)) * t_1;
	} else if (t_3 <= -0.2) {
		tmp = t_2 * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0))));
	} else if (t_3 <= 2e-7) {
		tmp = (t_4 / hypot(t_4, sin(kx))) * sin(th);
	} else if (t_3 <= 0.995) {
		tmp = t_2 * th;
	} else {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = t_1 / Math.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 t_4 = Math.abs(ky) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (Math.sin(th) / Math.hypot((kx * (1.0 + (-0.16666666666666666 * Math.pow(kx, 2.0)))), t_1)) * t_1;
	} else if (t_3 <= -0.2) {
		tmp = t_2 * (th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0))));
	} else if (t_3 <= 2e-7) {
		tmp = (t_4 / Math.hypot(t_4, Math.sin(kx))) * Math.sin(th);
	} else if (t_3 <= 0.995) {
		tmp = t_2 * th;
	} else {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = t_1 / math.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)))
	t_4 = math.fabs(ky) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(ky), 2.0)))
	tmp = 0
	if t_3 <= -0.98:
		tmp = (math.sin(th) / math.hypot((kx * (1.0 + (-0.16666666666666666 * math.pow(kx, 2.0)))), t_1)) * t_1
	elif t_3 <= -0.2:
		tmp = t_2 * (th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0))))
	elif t_3 <= 2e-7:
		tmp = (t_4 / math.hypot(t_4, math.sin(kx))) * math.sin(th)
	elif t_3 <= 0.995:
		tmp = t_2 * th
	else:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(t_1 / hypot(t_1, sin(kx)))
	t_3 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_4 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(Float64(sin(th) / hypot(Float64(kx * Float64(1.0 + Float64(-0.16666666666666666 * (kx ^ 2.0)))), t_1)) * t_1);
	elseif (t_3 <= -0.2)
		tmp = Float64(t_2 * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))));
	elseif (t_3 <= 2e-7)
		tmp = Float64(Float64(t_4 / hypot(t_4, sin(kx))) * sin(th));
	elseif (t_3 <= 0.995)
		tmp = Float64(t_2 * th);
	else
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 / hypot(t_1, sin(kx));
	t_3 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_4 = abs(ky) * (1.0 + (-0.16666666666666666 * (abs(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -0.98)
		tmp = (sin(th) / hypot((kx * (1.0 + (-0.16666666666666666 * (kx ^ 2.0)))), t_1)) * t_1;
	elseif (t_3 <= -0.2)
		tmp = t_2 * (th * (1.0 + (-0.16666666666666666 * (th ^ 2.0))));
	elseif (t_3 <= 2e-7)
		tmp = (t_4 / hypot(t_4, sin(kx))) * sin(th);
	elseif (t_3 <= 0.995)
		tmp = t_2 * th;
	else
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[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]}, Block[{t$95$4 = N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.98], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(kx * N[(1.0 + N[(-0.16666666666666666 * N[Power[kx, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(t$95$2 * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-7], N[(N[(t$95$4 / N[Sqrt[t$95$4 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], N[(t$95$2 * th), $MachinePrecision], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 57.7%

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

   6. Applied rewrites 57.7%

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

   if -0.97999999999999998 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 49.9%

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

   6. Applied rewrites 49.9%

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

   if -0.20000000000000001 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 50.9%

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

   6. Applied rewrites 50.9%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 55.2%

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

   9. Applied rewrites 55.2%

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

   if 1.9999999999999999e-7 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 50.2%

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

   if 0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

Alternative 4: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ t_3 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)}\\ t_4 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_5 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.98:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;t\_3 \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_5}{\mathsf{hypot}\left(t\_5, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_3 \cdot th\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (* (/ t_1 (hypot t_1 kx)) (sin th)))
        (t_3 (/ t_1 (hypot t_1 (sin kx))))
        (t_4 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_5
         (* (fabs ky) (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_4 -0.98)
      t_2
      (if (<= t_4 -0.2)
        (* t_3 (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0)))))
        (if (<= t_4 2e-7)
          (* (/ t_5 (hypot t_5 (sin kx))) (sin th))
          (if (<= t_4 0.995) (* t_3 th) t_2)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = (t_1 / hypot(t_1, kx)) * sin(th);
	double t_3 = t_1 / hypot(t_1, sin(kx));
	double t_4 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_5 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double tmp;
	if (t_4 <= -0.98) {
		tmp = t_2;
	} else if (t_4 <= -0.2) {
		tmp = t_3 * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0))));
	} else if (t_4 <= 2e-7) {
		tmp = (t_5 / hypot(t_5, sin(kx))) * sin(th);
	} else if (t_4 <= 0.995) {
		tmp = t_3 * th;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	double t_3 = t_1 / Math.hypot(t_1, Math.sin(kx));
	double t_4 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_5 = Math.abs(ky) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(ky), 2.0)));
	double tmp;
	if (t_4 <= -0.98) {
		tmp = t_2;
	} else if (t_4 <= -0.2) {
		tmp = t_3 * (th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0))));
	} else if (t_4 <= 2e-7) {
		tmp = (t_5 / Math.hypot(t_5, Math.sin(kx))) * Math.sin(th);
	} else if (t_4 <= 0.995) {
		tmp = t_3 * th;
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	t_3 = t_1 / math.hypot(t_1, math.sin(kx))
	t_4 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_5 = math.fabs(ky) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(ky), 2.0)))
	tmp = 0
	if t_4 <= -0.98:
		tmp = t_2
	elif t_4 <= -0.2:
		tmp = t_3 * (th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0))))
	elif t_4 <= 2e-7:
		tmp = (t_5 / math.hypot(t_5, math.sin(kx))) * math.sin(th)
	elif t_4 <= 0.995:
		tmp = t_3 * th
	else:
		tmp = t_2
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th))
	t_3 = Float64(t_1 / hypot(t_1, sin(kx)))
	t_4 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_5 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.98)
		tmp = t_2;
	elseif (t_4 <= -0.2)
		tmp = Float64(t_3 * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))));
	elseif (t_4 <= 2e-7)
		tmp = Float64(Float64(t_5 / hypot(t_5, sin(kx))) * sin(th));
	elseif (t_4 <= 0.995)
		tmp = Float64(t_3 * th);
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = (t_1 / hypot(t_1, kx)) * sin(th);
	t_3 = t_1 / hypot(t_1, sin(kx));
	t_4 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_5 = abs(ky) * (1.0 + (-0.16666666666666666 * (abs(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_4 <= -0.98)
		tmp = t_2;
	elseif (t_4 <= -0.2)
		tmp = t_3 * (th * (1.0 + (-0.16666666666666666 * (th ^ 2.0))));
	elseif (t_4 <= 2e-7)
		tmp = (t_5 / hypot(t_5, sin(kx))) * sin(th);
	elseif (t_4 <= 0.995)
		tmp = t_3 * th;
	else
		tmp = t_2;
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, -0.98], t$95$2, If[LessEqual[t$95$4, -0.2], N[(t$95$3 * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e-7], N[(N[(t$95$5 / N[Sqrt[t$95$5 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], N[(t$95$3 * th), $MachinePrecision], t$95$2]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.97999999999999998 or 0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

   if -0.97999999999999998 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 49.9%

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

   6. Applied rewrites 49.9%

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

   if -0.20000000000000001 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 50.9%

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

   6. Applied rewrites 50.9%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 55.2%

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

   9. Applied rewrites 55.2%

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

   if 1.9999999999999999e-7 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 50.2%

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

Alternative 5: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := {t\_2}^{2}\\ t_4 := \frac{t\_2}{\sqrt{{\sin kx}^{2} + t\_3}}\\ t_5 := \frac{t\_2}{\mathsf{hypot}\left(t\_2, \sin kx\right)} \cdot th\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.995:\\ \;\;\;\;\frac{t\_2}{\sqrt{{kx}^{2} + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(t\_2, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (* (fabs ky) (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0)))))
        (t_2 (sin (fabs ky)))
        (t_3 (pow t_2 2.0))
        (t_4 (/ t_2 (sqrt (+ (pow (sin kx) 2.0) t_3))))
        (t_5 (* (/ t_2 (hypot t_2 (sin kx))) th)))
   (*
    (copysign 1.0 ky)
    (if (<= t_4 -0.995)
      (* (/ t_2 (sqrt (+ (pow kx 2.0) t_3))) (sin th))
      (if (<= t_4 -0.2)
        t_5
        (if (<= t_4 2e-7)
          (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
          (if (<= t_4 0.995) t_5 (* (/ t_2 (hypot t_2 kx)) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double t_2 = sin(fabs(ky));
	double t_3 = pow(t_2, 2.0);
	double t_4 = t_2 / sqrt((pow(sin(kx), 2.0) + t_3));
	double t_5 = (t_2 / hypot(t_2, sin(kx))) * th;
	double tmp;
	if (t_4 <= -0.995) {
		tmp = (t_2 / sqrt((pow(kx, 2.0) + t_3))) * sin(th);
	} else if (t_4 <= -0.2) {
		tmp = t_5;
	} else if (t_4 <= 2e-7) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else if (t_4 <= 0.995) {
		tmp = t_5;
	} else {
		tmp = (t_2 / hypot(t_2, kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(ky) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(ky), 2.0)));
	double t_2 = Math.sin(Math.abs(ky));
	double t_3 = Math.pow(t_2, 2.0);
	double t_4 = t_2 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_3));
	double t_5 = (t_2 / Math.hypot(t_2, Math.sin(kx))) * th;
	double tmp;
	if (t_4 <= -0.995) {
		tmp = (t_2 / Math.sqrt((Math.pow(kx, 2.0) + t_3))) * Math.sin(th);
	} else if (t_4 <= -0.2) {
		tmp = t_5;
	} else if (t_4 <= 2e-7) {
		tmp = (t_1 / Math.hypot(t_1, Math.sin(kx))) * Math.sin(th);
	} else if (t_4 <= 0.995) {
		tmp = t_5;
	} else {
		tmp = (t_2 / Math.hypot(t_2, kx)) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(ky) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(ky), 2.0)))
	t_2 = math.sin(math.fabs(ky))
	t_3 = math.pow(t_2, 2.0)
	t_4 = t_2 / math.sqrt((math.pow(math.sin(kx), 2.0) + t_3))
	t_5 = (t_2 / math.hypot(t_2, math.sin(kx))) * th
	tmp = 0
	if t_4 <= -0.995:
		tmp = (t_2 / math.sqrt((math.pow(kx, 2.0) + t_3))) * math.sin(th)
	elif t_4 <= -0.2:
		tmp = t_5
	elif t_4 <= 2e-7:
		tmp = (t_1 / math.hypot(t_1, math.sin(kx))) * math.sin(th)
	elif t_4 <= 0.995:
		tmp = t_5
	else:
		tmp = (t_2 / math.hypot(t_2, kx)) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	t_2 = sin(abs(ky))
	t_3 = t_2 ^ 2.0
	t_4 = Float64(t_2 / sqrt(Float64((sin(kx) ^ 2.0) + t_3)))
	t_5 = Float64(Float64(t_2 / hypot(t_2, sin(kx))) * th)
	tmp = 0.0
	if (t_4 <= -0.995)
		tmp = Float64(Float64(t_2 / sqrt(Float64((kx ^ 2.0) + t_3))) * sin(th));
	elseif (t_4 <= -0.2)
		tmp = t_5;
	elseif (t_4 <= 2e-7)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	elseif (t_4 <= 0.995)
		tmp = t_5;
	else
		tmp = Float64(Float64(t_2 / hypot(t_2, kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(ky) * (1.0 + (-0.16666666666666666 * (abs(ky) ^ 2.0)));
	t_2 = sin(abs(ky));
	t_3 = t_2 ^ 2.0;
	t_4 = t_2 / sqrt(((sin(kx) ^ 2.0) + t_3));
	t_5 = (t_2 / hypot(t_2, sin(kx))) * th;
	tmp = 0.0;
	if (t_4 <= -0.995)
		tmp = (t_2 / sqrt(((kx ^ 2.0) + t_3))) * sin(th);
	elseif (t_4 <= -0.2)
		tmp = t_5;
	elseif (t_4 <= 2e-7)
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	elseif (t_4 <= 0.995)
		tmp = t_5;
	else
		tmp = (t_2 / hypot(t_2, kx)) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 / N[Sqrt[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, -0.995], N[(N[(t$95$2 / N[Sqrt[N[(N[Power[kx, 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.2], t$95$5, If[LessEqual[t$95$4, 2e-7], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], t$95$5, N[(N[(t$95$2 / N[Sqrt[t$95$2 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.994999999999999996

   1. Initial program 93.6%

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

   2. Taylor expanded in kx around 0

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

   4. Applied rewrites 51.9%

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

   if -0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1.9999999999999999e-7 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 50.2%

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

   if -0.20000000000000001 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 50.9%

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

   6. Applied rewrites 50.9%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 55.2%

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

   9. Applied rewrites 55.2%

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

   if 0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

Alternative 6: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot th\\ t_4 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ t_5 := \left|ky\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|ky\right|\right)}^{2}\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.995:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_5}{\mathsf{hypot}\left(t\_5, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

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 (hypot t_1 (sin kx))) th))
        (t_4 (* (/ t_1 (hypot t_1 kx)) (sin th)))
        (t_5
         (* (fabs ky) (+ 1.0 (* -0.16666666666666666 (pow (fabs ky) 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -0.995)
      t_4
      (if (<= t_2 -0.2)
        t_3
        (if (<= t_2 2e-7)
          (* (/ t_5 (hypot t_5 (sin kx))) (sin th))
          (if (<= t_2 0.995) t_3 t_4)))))))

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 / hypot(t_1, sin(kx))) * th;
	double t_4 = (t_1 / hypot(t_1, kx)) * sin(th);
	double t_5 = fabs(ky) * (1.0 + (-0.16666666666666666 * pow(fabs(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.995) {
		tmp = t_4;
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 2e-7) {
		tmp = (t_5 / hypot(t_5, sin(kx))) * sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_3 = (t_1 / Math.hypot(t_1, Math.sin(kx))) * th;
	double t_4 = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	double t_5 = Math.abs(ky) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.995) {
		tmp = t_4;
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 2e-7) {
		tmp = (t_5 / Math.hypot(t_5, Math.sin(kx))) * Math.sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_3 = (t_1 / math.hypot(t_1, math.sin(kx))) * th
	t_4 = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	t_5 = math.fabs(ky) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.995:
		tmp = t_4
	elif t_2 <= -0.2:
		tmp = t_3
	elif t_2 <= 2e-7:
		tmp = (t_5 / math.hypot(t_5, math.sin(kx))) * math.sin(th)
	elif t_2 <= 0.995:
		tmp = t_3
	else:
		tmp = t_4
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_3 = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * th)
	t_4 = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th))
	t_5 = Float64(abs(ky) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.995)
		tmp = t_4;
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 2e-7)
		tmp = Float64(Float64(t_5 / hypot(t_5, sin(kx))) * sin(th));
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_3 = (t_1 / hypot(t_1, sin(kx))) * th;
	t_4 = (t_1 / hypot(t_1, kx)) * sin(th);
	t_5 = abs(ky) * (1.0 + (-0.16666666666666666 * (abs(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.995)
		tmp = t_4;
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 2e-7)
		tmp = (t_5 / hypot(t_5, sin(kx))) * sin(th);
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = t_4;
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(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 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[ky], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -0.995], t$95$4, If[LessEqual[t$95$2, -0.2], t$95$3, If[LessEqual[t$95$2, 2e-7], N[(N[(t$95$5 / N[Sqrt[t$95$5 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], t$95$3, t$95$4]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.994999999999999996 or 0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

   if -0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1.9999999999999999e-7 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 50.2%

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

   if -0.20000000000000001 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 50.9%

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

   6. Applied rewrites 50.9%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 55.2%

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

   9. Applied rewrites 55.2%

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

Alternative 7: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ t_3 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_4 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot th\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.995:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)}{\left|ky\right|}}\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (* (/ t_1 (hypot t_1 kx)) (sin th)))
        (t_3 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_4 (* (/ t_1 (hypot t_1 (sin kx))) th)))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.995)
      t_2
      (if (<= t_3 -0.2)
        t_4
        (if (<= t_3 2e-7)
          (/ (sin th) (/ (hypot (sin kx) (fabs ky)) (fabs ky)))
          (if (<= t_3 0.995) t_4 t_2)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = (t_1 / hypot(t_1, kx)) * sin(th);
	double t_3 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_4 = (t_1 / hypot(t_1, sin(kx))) * th;
	double tmp;
	if (t_3 <= -0.995) {
		tmp = t_2;
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 2e-7) {
		tmp = sin(th) / (hypot(sin(kx), fabs(ky)) / fabs(ky));
	} else if (t_3 <= 0.995) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	double t_3 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_4 = (t_1 / Math.hypot(t_1, Math.sin(kx))) * th;
	double tmp;
	if (t_3 <= -0.995) {
		tmp = t_2;
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 2e-7) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), Math.abs(ky)) / Math.abs(ky));
	} else if (t_3 <= 0.995) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	t_3 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_4 = (t_1 / math.hypot(t_1, math.sin(kx))) * th
	tmp = 0
	if t_3 <= -0.995:
		tmp = t_2
	elif t_3 <= -0.2:
		tmp = t_4
	elif t_3 <= 2e-7:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), math.fabs(ky)) / math.fabs(ky))
	elif t_3 <= 0.995:
		tmp = t_4
	else:
		tmp = t_2
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th))
	t_3 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_4 = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * th)
	tmp = 0.0
	if (t_3 <= -0.995)
		tmp = t_2;
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 2e-7)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), abs(ky)) / abs(ky)));
	elseif (t_3 <= 0.995)
		tmp = t_4;
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = (t_1 / hypot(t_1, kx)) * sin(th);
	t_3 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_4 = (t_1 / hypot(t_1, sin(kx))) * th;
	tmp = 0.0;
	if (t_3 <= -0.995)
		tmp = t_2;
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 2e-7)
		tmp = sin(th) / (hypot(sin(kx), abs(ky)) / abs(ky));
	elseif (t_3 <= 0.995)
		tmp = t_4;
	else
		tmp = t_2;
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.995], t$95$2, If[LessEqual[t$95$3, -0.2], t$95$4, If[LessEqual[t$95$3, 2e-7], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], t$95$4, t$95$2]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.994999999999999996 or 0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

   if -0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1.9999999999999999e-7 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 50.2%

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

   if -0.20000000000000001 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.64 N/A

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

      11. pow2 N/A

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

      12. lift-hypot.f64 N/A

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

      13. div-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. div-flip-rev N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 52.5%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 64.7%

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

Alternative 8: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ t_3 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_4 := \frac{th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.995:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)}{\left|ky\right|}}\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (* (/ t_1 (hypot t_1 kx)) (sin th)))
        (t_3 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_4 (* (/ th (hypot (sin kx) t_1)) t_1)))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.995)
      t_2
      (if (<= t_3 -0.2)
        t_4
        (if (<= t_3 2e-7)
          (/ (sin th) (/ (hypot (sin kx) (fabs ky)) (fabs ky)))
          (if (<= t_3 0.995) t_4 t_2)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = (t_1 / hypot(t_1, kx)) * sin(th);
	double t_3 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_4 = (th / hypot(sin(kx), t_1)) * t_1;
	double tmp;
	if (t_3 <= -0.995) {
		tmp = t_2;
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 2e-7) {
		tmp = sin(th) / (hypot(sin(kx), fabs(ky)) / fabs(ky));
	} else if (t_3 <= 0.995) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	double t_3 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_4 = (th / Math.hypot(Math.sin(kx), t_1)) * t_1;
	double tmp;
	if (t_3 <= -0.995) {
		tmp = t_2;
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 2e-7) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), Math.abs(ky)) / Math.abs(ky));
	} else if (t_3 <= 0.995) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	t_3 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_4 = (th / math.hypot(math.sin(kx), t_1)) * t_1
	tmp = 0
	if t_3 <= -0.995:
		tmp = t_2
	elif t_3 <= -0.2:
		tmp = t_4
	elif t_3 <= 2e-7:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), math.fabs(ky)) / math.fabs(ky))
	elif t_3 <= 0.995:
		tmp = t_4
	else:
		tmp = t_2
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th))
	t_3 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_4 = Float64(Float64(th / hypot(sin(kx), t_1)) * t_1)
	tmp = 0.0
	if (t_3 <= -0.995)
		tmp = t_2;
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 2e-7)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), abs(ky)) / abs(ky)));
	elseif (t_3 <= 0.995)
		tmp = t_4;
	else
		tmp = t_2;
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = (t_1 / hypot(t_1, kx)) * sin(th);
	t_3 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_4 = (th / hypot(sin(kx), t_1)) * t_1;
	tmp = 0.0;
	if (t_3 <= -0.995)
		tmp = t_2;
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 2e-7)
		tmp = sin(th) / (hypot(sin(kx), abs(ky)) / abs(ky));
	elseif (t_3 <= 0.995)
		tmp = t_4;
	else
		tmp = t_2;
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.995], t$95$2, If[LessEqual[t$95$3, -0.2], t$95$4, If[LessEqual[t$95$3, 2e-7], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], t$95$4, t$95$2]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.994999999999999996 or 0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

   if -0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1.9999999999999999e-7 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.2%

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

   if -0.20000000000000001 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.64 N/A

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

      11. pow2 N/A

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

      12. lift-hypot.f64 N/A

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

      13. div-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. div-flip-rev N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 52.5%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 64.7%

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

Alternative 9: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)}{\left|ky\right|}}\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \frac{t\_2}{\sqrt{{\sin kx}^{2} + {t\_2}^{2}}}\\ t_4 := \frac{th}{\mathsf{hypot}\left(\sin kx, t\_2\right)} \cdot t\_2\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.995:\\ \;\;\;\;\frac{t\_2 \cdot \sin th}{\mathsf{hypot}\left(kx, t\_2\right)}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.9999999823355598:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (/ (hypot (sin kx) (fabs ky)) (fabs ky))))
        (t_2 (sin (fabs ky)))
        (t_3 (/ t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_2 2.0)))))
        (t_4 (* (/ th (hypot (sin kx) t_2)) t_2)))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.995)
      (/ (* t_2 (sin th)) (hypot kx t_2))
      (if (<= t_3 -0.2)
        t_4
        (if (<= t_3 2e-7) t_1 (if (<= t_3 0.9999999823355598) t_4 t_1)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / (hypot(sin(kx), fabs(ky)) / fabs(ky));
	double t_2 = sin(fabs(ky));
	double t_3 = t_2 / sqrt((pow(sin(kx), 2.0) + pow(t_2, 2.0)));
	double t_4 = (th / hypot(sin(kx), t_2)) * t_2;
	double tmp;
	if (t_3 <= -0.995) {
		tmp = (t_2 * sin(th)) / hypot(kx, t_2);
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 2e-7) {
		tmp = t_1;
	} else if (t_3 <= 0.9999999823355598) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / (Math.hypot(Math.sin(kx), Math.abs(ky)) / Math.abs(ky));
	double t_2 = Math.sin(Math.abs(ky));
	double t_3 = t_2 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_2, 2.0)));
	double t_4 = (th / Math.hypot(Math.sin(kx), t_2)) * t_2;
	double tmp;
	if (t_3 <= -0.995) {
		tmp = (t_2 * Math.sin(th)) / Math.hypot(kx, t_2);
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 2e-7) {
		tmp = t_1;
	} else if (t_3 <= 0.9999999823355598) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / (math.hypot(math.sin(kx), math.fabs(ky)) / math.fabs(ky))
	t_2 = math.sin(math.fabs(ky))
	t_3 = t_2 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_2, 2.0)))
	t_4 = (th / math.hypot(math.sin(kx), t_2)) * t_2
	tmp = 0
	if t_3 <= -0.995:
		tmp = (t_2 * math.sin(th)) / math.hypot(kx, t_2)
	elif t_3 <= -0.2:
		tmp = t_4
	elif t_3 <= 2e-7:
		tmp = t_1
	elif t_3 <= 0.9999999823355598:
		tmp = t_4
	else:
		tmp = t_1
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / Float64(hypot(sin(kx), abs(ky)) / abs(ky)))
	t_2 = sin(abs(ky))
	t_3 = Float64(t_2 / sqrt(Float64((sin(kx) ^ 2.0) + (t_2 ^ 2.0))))
	t_4 = Float64(Float64(th / hypot(sin(kx), t_2)) * t_2)
	tmp = 0.0
	if (t_3 <= -0.995)
		tmp = Float64(Float64(t_2 * sin(th)) / hypot(kx, t_2));
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 2e-7)
		tmp = t_1;
	elseif (t_3 <= 0.9999999823355598)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / (hypot(sin(kx), abs(ky)) / abs(ky));
	t_2 = sin(abs(ky));
	t_3 = t_2 / sqrt(((sin(kx) ^ 2.0) + (t_2 ^ 2.0)));
	t_4 = (th / hypot(sin(kx), t_2)) * t_2;
	tmp = 0.0;
	if (t_3 <= -0.995)
		tmp = (t_2 * sin(th)) / hypot(kx, t_2);
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 2e-7)
		tmp = t_1;
	elseif (t_3 <= 0.9999999823355598)
		tmp = t_4;
	else
		tmp = t_1;
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.995], N[(N[(t$95$2 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], t$95$4, If[LessEqual[t$95$3, 2e-7], t$95$1, If[LessEqual[t$95$3, 0.9999999823355598], t$95$4, t$95$1]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.994999999999999996

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 54.3%

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-hypot.f64 54.3%

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

   8. Applied rewrites 54.3%

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

   if -0.994999999999999996 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1.9999999999999999e-7 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 0.999999982335559756

   1. Initial program 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.2%

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

   if -0.20000000000000001 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7 or 0.999999982335559756 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.64 N/A

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

      11. pow2 N/A

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

      12. lift-hypot.f64 N/A

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

      13. div-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. div-flip-rev N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 52.5%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 64.7%

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

Alternative 10: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq -0.2:\\ \;\;\;\;\frac{\sin th \cdot t\_1}{\sqrt{\left(1 - \cos \left(\left|ky\right| + \left|ky\right|\right)\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)}{\left|ky\right|}}\\ \end{array} \end{array} \]

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)
      (/ (* (sin th) t_1) (sqrt (* (- 1.0 (cos (+ (fabs ky) (fabs ky)))) 0.5)))
      (/ (sin th) (/ (hypot (sin kx) (fabs ky)) (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 = (sin(th) * t_1) / sqrt(((1.0 - cos((fabs(ky) + fabs(ky)))) * 0.5));
	} else {
		tmp = sin(th) / (hypot(sin(kx), fabs(ky)) / 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 = (Math.sin(th) * t_1) / Math.sqrt(((1.0 - Math.cos((Math.abs(ky) + Math.abs(ky)))) * 0.5));
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), Math.abs(ky)) / 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 = (math.sin(th) * t_1) / math.sqrt(((1.0 - math.cos((math.fabs(ky) + math.fabs(ky)))) * 0.5))
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), math.fabs(ky)) / 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(sin(th) * t_1) / sqrt(Float64(Float64(1.0 - cos(Float64(abs(ky) + abs(ky)))) * 0.5)));
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), abs(ky)) / 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 = (sin(th) * t_1) / sqrt(((1.0 - cos((abs(ky) + abs(ky)))) * 0.5));
	else
		tmp = sin(th) / (hypot(sin(kx), abs(ky)) / 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[(N[Sin[th], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(N[Abs[ky], $MachinePrecision] + N[Abs[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 93.6%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.64 N/A

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

      2. lower-sin.64 40.2%

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

   4. Applied rewrites 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 40.8%

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

      7. lift-pow.64 N/A

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

      8. pow2 N/A

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

      9. lift-sin.64 N/A

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

      10. lift-sin.64 N/A

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

      11. sin-mult N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 30.6%

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

   if -0.20000000000000001 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.64 N/A

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

      11. pow2 N/A

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

      12. lift-hypot.f64 N/A

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

      13. div-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. div-flip-rev N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 52.5%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 64.7%

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

Alternative 11: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq -0.2:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\sqrt{\left(1 - \cos \left(\left|ky\right| + \left|ky\right|\right)\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)}{\left|ky\right|}}\\ \end{array} \end{array} \]

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)
      (* t_1 (/ (sin th) (sqrt (* (- 1.0 (cos (+ (fabs ky) (fabs ky)))) 0.5))))
      (/ (sin th) (/ (hypot (sin kx) (fabs ky)) (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 = t_1 * (sin(th) / sqrt(((1.0 - cos((fabs(ky) + fabs(ky)))) * 0.5)));
	} else {
		tmp = sin(th) / (hypot(sin(kx), fabs(ky)) / 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 = t_1 * (Math.sin(th) / Math.sqrt(((1.0 - Math.cos((Math.abs(ky) + Math.abs(ky)))) * 0.5)));
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), Math.abs(ky)) / 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 = t_1 * (math.sin(th) / math.sqrt(((1.0 - math.cos((math.fabs(ky) + math.fabs(ky)))) * 0.5)))
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), math.fabs(ky)) / 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(t_1 * Float64(sin(th) / sqrt(Float64(Float64(1.0 - cos(Float64(abs(ky) + abs(ky)))) * 0.5))));
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), abs(ky)) / 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 = t_1 * (sin(th) / sqrt(((1.0 - cos((abs(ky) + abs(ky)))) * 0.5)));
	else
		tmp = sin(th) / (hypot(sin(kx), abs(ky)) / 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[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(N[Abs[ky], $MachinePrecision] + N[Abs[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 93.6%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.64 N/A

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

      2. lower-sin.64 40.2%

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

   4. Applied rewrites 40.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 40.2%

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

      7. lift-pow.64 N/A

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

      8. pow2 N/A

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

      9. lift-sin.64 N/A

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

      10. lift-sin.64 N/A

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

      11. sin-mult N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 30.8%

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

   if -0.20000000000000001 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.64 N/A

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

      11. pow2 N/A

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

      12. lift-hypot.f64 N/A

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

      13. div-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. div-flip-rev N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 52.5%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 64.7%

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

Alternative 12: 71.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\frac{t\_1}{\sqrt{{t\_1}^{2}}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)}{\left|ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_1 -0.02)
      (* (/ t_1 (sqrt (pow t_1 2.0))) th)
      (/ (sin th) (/ (hypot (sin kx) (fabs ky)) (fabs ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = (t_1 / sqrt(pow(t_1, 2.0))) * th;
	} else {
		tmp = sin(th) / (hypot(sin(kx), fabs(ky)) / 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 <= -0.02) {
		tmp = (t_1 / Math.sqrt(Math.pow(t_1, 2.0))) * th;
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), Math.abs(ky)) / 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 <= -0.02:
		tmp = (t_1 / math.sqrt(math.pow(t_1, 2.0))) * th
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), math.fabs(ky)) / 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 <= -0.02)
		tmp = Float64(Float64(t_1 / sqrt((t_1 ^ 2.0))) * th);
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), abs(ky)) / 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 <= -0.02)
		tmp = (t_1 / sqrt((t_1 ^ 2.0))) * th;
	else
		tmp = sin(th) / (hypot(sin(kx), abs(ky)) / 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, -0.02], N[(N[(t$95$1 / N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.64 ky) < -0.0200000000000000004

   1. Initial program 93.6%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.64 N/A

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

      2. lower-sin.64 40.2%

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

   4. Applied rewrites 40.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 20.6%

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

   if -0.0200000000000000004 < (sin.64 ky)

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.64 N/A

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

      11. pow2 N/A

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

      12. lift-hypot.f64 N/A

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

      13. div-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. div-flip-rev N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 52.5%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 64.7%

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

Alternative 13: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \left|ky\right|\right)}{\left|ky\right|}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_1 -0.02)
      (* (/ t_1 (hypot t_1 kx)) th)
      (/ (sin th) (/ (hypot (sin kx) (fabs ky)) (fabs ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = (t_1 / hypot(t_1, kx)) * th;
	} else {
		tmp = sin(th) / (hypot(sin(kx), fabs(ky)) / 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 <= -0.02) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * th;
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), Math.abs(ky)) / 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 <= -0.02:
		tmp = (t_1 / math.hypot(t_1, kx)) * th
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), math.fabs(ky)) / 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 <= -0.02)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * th);
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), abs(ky)) / 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 <= -0.02)
		tmp = (t_1 / hypot(t_1, kx)) * th;
	else
		tmp = sin(th) / (hypot(sin(kx), abs(ky)) / 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, -0.02], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.64 ky) < -0.0200000000000000004

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 33.0%

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

   if -0.0200000000000000004 < (sin.64 ky)

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.64 N/A

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

      11. pow2 N/A

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

      12. lift-hypot.f64 N/A

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

      13. div-flip-rev N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

      17. div-flip-rev N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 52.5%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 64.7%

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

Alternative 14: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_1 -0.02)
      (* (/ t_1 (hypot t_1 kx)) th)
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = (t_1 / hypot(t_1, kx)) * th;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * th;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if t_1 <= -0.02:
		tmp = (t_1 / math.hypot(t_1, kx)) * th
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * th);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if (t_1 <= -0.02)
		tmp = (t_1 / hypot(t_1, kx)) * th;
	else
		tmp = (abs(ky) / hypot(abs(ky), sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -0.02], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * th), $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.64 ky) < -0.0200000000000000004

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 33.0%

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

   if -0.0200000000000000004 < (sin.64 ky)

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 51.2%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.7%

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

Alternative 15: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. lift-sqrt.64 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.64 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.64 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 51.2%

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

7. Taylor expanded in ky around 0

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

9. Applied rewrites 64.7%

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

Alternative 16: 63.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 0.01)
      (* (sin th) (/ (fabs ky) (fabs (sin kx))))
      (* (/ (fabs ky) (hypot (fabs ky) kx)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 0.01) {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 0.01) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(Math.sin(kx)));
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), kx)) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 0.01:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(math.sin(kx)))
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), kx)) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.01)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.01)
		tmp = sin(th) * (abs(ky) / abs(sin(kx)));
	else
		tmp = (abs(ky) / hypot(abs(ky), kx)) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 93.6%

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

   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.64 N/A

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

      3. lower-pow.64 N/A

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

      4. lower-sin.64 36.2%

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

   4. Applied rewrites 36.2%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   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 36.2%

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

      4. lift-sqrt.64 N/A

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

      5. lift-pow.64 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.64 39.3%

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

   6. Applied rewrites 39.3%

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

   if 0.0100000000000000002 < (/.f64 (sin.64 ky) (sqrt.64 (+.f64 (pow.64 (sin.64 kx) #s(literal 2 binary64)) (pow.64 (sin.64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 34.1%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 46.3%

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

Alternative 17: 49.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := {\sin kx}^{2}\\ \mathbf{if}\;t\_1 \leq 0.01886:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{t\_1}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0)))
   (if (<= t_1 0.01886)
     (* (/ ky (hypot ky kx)) (sin th))
     (* (/ ky (sqrt t_1)) th))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if (t_1 <= 0.01886) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = (ky / sqrt(t_1)) * th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 2.0);
	double tmp;
	if (t_1 <= 0.01886) {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	} else {
		tmp = (ky / Math.sqrt(t_1)) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	tmp = 0
	if t_1 <= 0.01886:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	else:
		tmp = (ky / math.sqrt(t_1)) * th
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (t_1 <= 0.01886)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt(t_1)) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	tmp = 0.0;
	if (t_1 <= 0.01886)
		tmp = (ky / hypot(ky, kx)) * sin(th);
	else
		tmp = (ky / sqrt(t_1)) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 0.01886], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.64 (sin.64 kx) #s(literal 2 binary64)) < 0.0188599999999999983

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 34.1%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 46.3%

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

   if 0.0188599999999999983 < (pow.64 (sin.64 kx) #s(literal 2 binary64))

   1. Initial program 93.6%

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

   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.64 N/A

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

      3. lower-pow.64 N/A

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

      4. lower-sin.64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 18.9%

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

Alternative 18: 49.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.01886:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sqrt{1 - \cos \left(kx + kx\right)}}{\sqrt{2}}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 0.01886)
   (* (/ ky (hypot ky kx)) (sin th))
   (* (/ ky (/ (sqrt (- 1.0 (cos (+ kx kx)))) (sqrt 2.0))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 0.01886) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = (ky / (sqrt((1.0 - cos((kx + kx)))) / sqrt(2.0))) * th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(kx), 2.0) <= 0.01886) {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	} else {
		tmp = (ky / (Math.sqrt((1.0 - Math.cos((kx + kx)))) / Math.sqrt(2.0))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(kx), 2.0) <= 0.01886:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	else:
		tmp = (ky / (math.sqrt((1.0 - math.cos((kx + kx)))) / math.sqrt(2.0))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 0.01886)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(Float64(ky / Float64(sqrt(Float64(1.0 - cos(Float64(kx + kx)))) / sqrt(2.0))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(kx) ^ 2.0) <= 0.01886)
		tmp = (ky / hypot(ky, kx)) * sin(th);
	else
		tmp = (ky / (sqrt((1.0 - cos((kx + kx)))) / sqrt(2.0))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 0.01886], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.64 (sin.64 kx) #s(literal 2 binary64)) < 0.0188599999999999983

   1. Initial program 93.6%

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

      1. lift-sqrt.64 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.64 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.64 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 58.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 34.1%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 46.3%

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

   if 0.0188599999999999983 < (pow.64 (sin.64 kx) #s(literal 2 binary64))

   1. Initial program 93.6%

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

   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.64 N/A

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

      3. lower-pow.64 N/A

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

      4. lower-sin.64 36.2%

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

   4. Applied rewrites 36.2%

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

      1. lift-sqrt.64 N/A

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

      2. lift-pow.64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.64 N/A

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

      5. lift-sin.64 N/A

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

      6. sin-mult N/A

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

      7. sqrt-div N/A

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

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

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

      9. lower-unsound-sqrt.64 N/A

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

      10. sub-to-mult N/A

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

      11. +-inverses N/A

          \[\leadsto \frac{ky}{\frac{\sqrt{\left(1 - \frac{\cos \left(kx + kx\right)}{\cos 0}\right) \cdot \cos \left(kx - kx\right)}}{\sqrt{2}}} \cdot \sin th \]

      12. cos-0 N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{ky}{\frac{\sqrt{\left(1 - \frac{\cos \left(kx + kx\right)}{1}\right) \cdot \cos 0}}{\sqrt{2}}} \cdot \sin th \]

      14. cos-0 N/A

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

      15. sub-to-mult N/A

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

      16. lower--.f64 N/A

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

      17. lower-cos.64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-unsound-sqrt.64 26.9%

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

   6. Applied rewrites 26.9%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 14.3%

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

Alternative 19: 26.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{\sin \left(\left|kx\right|\right)}^{2} \leq 0.005:\\ \;\;\;\;\frac{ky}{\left|kx\right|} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sqrt{1 - \cos \left(\left|kx\right| + \left|kx\right|\right)}}{\sqrt{2}}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin (fabs kx)) 2.0) 0.005)
   (* (/ ky (fabs kx)) (sin th))
   (* (/ ky (/ (sqrt (- 1.0 (cos (+ (fabs kx) (fabs kx))))) (sqrt 2.0))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(fabs(kx)), 2.0) <= 0.005) {
		tmp = (ky / fabs(kx)) * sin(th);
	} else {
		tmp = (ky / (sqrt((1.0 - cos((fabs(kx) + fabs(kx))))) / sqrt(2.0))) * th;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(abs(kx)) ** 2.0d0) <= 0.005d0) then
        tmp = (ky / abs(kx)) * sin(th)
    else
        tmp = (ky / (sqrt((1.0d0 - cos((abs(kx) + abs(kx))))) / sqrt(2.0d0))) * th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(Math.abs(kx)), 2.0) <= 0.005) {
		tmp = (ky / Math.abs(kx)) * Math.sin(th);
	} else {
		tmp = (ky / (Math.sqrt((1.0 - Math.cos((Math.abs(kx) + Math.abs(kx))))) / Math.sqrt(2.0))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(math.fabs(kx)), 2.0) <= 0.005:
		tmp = (ky / math.fabs(kx)) * math.sin(th)
	else:
		tmp = (ky / (math.sqrt((1.0 - math.cos((math.fabs(kx) + math.fabs(kx))))) / math.sqrt(2.0))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(abs(kx)) ^ 2.0) <= 0.005)
		tmp = Float64(Float64(ky / abs(kx)) * sin(th));
	else
		tmp = Float64(Float64(ky / Float64(sqrt(Float64(1.0 - cos(Float64(abs(kx) + abs(kx))))) / sqrt(2.0))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(abs(kx)) ^ 2.0) <= 0.005)
		tmp = (ky / abs(kx)) * sin(th);
	else
		tmp = (ky / (sqrt((1.0 - cos((abs(kx) + abs(kx))))) / sqrt(2.0))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], 0.005], N[(N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[(N[Sqrt[N[(1.0 - N[Cos[N[(N[Abs[kx], $MachinePrecision] + N[Abs[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.64 (sin.64 kx) #s(literal 2 binary64)) < 0.0050000000000000001

   1. Initial program 93.6%

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

   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.64 N/A

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

      3. lower-pow.64 N/A

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

      4. lower-sin.64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.7%

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

   7. Applied rewrites 16.7%

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

   if 0.0050000000000000001 < (pow.64 (sin.64 kx) #s(literal 2 binary64))

   1. Initial program 93.6%

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

   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.64 N/A

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

      3. lower-pow.64 N/A

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

      4. lower-sin.64 36.2%

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

   4. Applied rewrites 36.2%

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

      1. lift-sqrt.64 N/A

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

      2. lift-pow.64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.64 N/A

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

      5. lift-sin.64 N/A

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

      6. sin-mult N/A

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

      7. sqrt-div N/A

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

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

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

      9. lower-unsound-sqrt.64 N/A

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

      10. sub-to-mult N/A

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

      11. +-inverses N/A

          \[\leadsto \frac{ky}{\frac{\sqrt{\left(1 - \frac{\cos \left(kx + kx\right)}{\cos 0}\right) \cdot \cos \left(kx - kx\right)}}{\sqrt{2}}} \cdot \sin th \]

      12. cos-0 N/A

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

      13. +-inverses N/A

          \[\leadsto \frac{ky}{\frac{\sqrt{\left(1 - \frac{\cos \left(kx + kx\right)}{1}\right) \cdot \cos 0}}{\sqrt{2}}} \cdot \sin th \]

      14. cos-0 N/A

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

      15. sub-to-mult N/A

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

      16. lower--.f64 N/A

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

      17. lower-cos.64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-unsound-sqrt.64 26.9%

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

   6. Applied rewrites 26.9%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 14.3%

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

Alternative 20: 22.2% 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 93.6%

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

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.64 N/A

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

   3. lower-pow.64 N/A

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

   4. lower-sin.64 36.2%

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

4. Applied rewrites 36.2%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 16.7%

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

7. Applied rewrites 16.7%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
8. Add Preprocessing

Alternative 21: 15.8% accurate, 14.4× speedup

Math

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

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ 1.0 (/ (fabs kx) ky)) th))

C

double code(double kx, double ky, double th) {
	return (1.0 / (fabs(kx) / ky)) * 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 = (1.0d0 / (abs(kx) / ky)) * th
end function

Java

public static double code(double kx, double ky, double th) {
	return (1.0 / (Math.abs(kx) / ky)) * th;
}

Python

def code(kx, ky, th):
	return (1.0 / (math.fabs(kx) / ky)) * th

Julia

function code(kx, ky, th)
	return Float64(Float64(1.0 / Float64(abs(kx) / ky)) * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (1.0 / (abs(kx) / ky)) * th;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(1.0 / N[(N[Abs[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

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.64 N/A

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

   3. lower-pow.64 N/A

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

   4. lower-sin.64 36.2%

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

4. Applied rewrites 36.2%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 16.7%

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

7. Applied rewrites 16.7%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 13.7%

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

    1. lift-/.f64 N/A

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

    2. div-flip N/A

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

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

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

    4. lower-unsound-/.f64 13.7%

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

12. Applied rewrites 13.7%

    \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot th \]
13. Add Preprocessing

Alternative 22: 15.8% accurate, 14.9× speedup

Math

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

FPCore

(FPCore (kx ky th) :precision binary64 (* (* (/ 1.0 (fabs kx)) ky) th))

C

double code(double kx, double ky, double th) {
	return ((1.0 / fabs(kx)) * ky) * 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 = ((1.0d0 / abs(kx)) * ky) * th
end function

Java

public static double code(double kx, double ky, double th) {
	return ((1.0 / Math.abs(kx)) * ky) * th;
}

Python

def code(kx, ky, th):
	return ((1.0 / math.fabs(kx)) * ky) * th

Julia

function code(kx, ky, th)
	return Float64(Float64(Float64(1.0 / abs(kx)) * ky) * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = ((1.0 / abs(kx)) * ky) * th;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[(1.0 / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

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.64 N/A

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

   3. lower-pow.64 N/A

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

   4. lower-sin.64 36.2%

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

4. Applied rewrites 36.2%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 16.7%

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

7. Applied rewrites 16.7%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 13.7%

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

    1. lift-/.f64 N/A

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

    2. mult-flip N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower-/.f64 13.7%

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

12. Applied rewrites 13.7%

    \[\leadsto \left(\frac{1}{kx} \cdot ky\right) \cdot th \]
13. Add Preprocessing

Alternative 23: 15.8% accurate, 20.0× speedup

Math

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

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ ky (fabs kx)) th))

C

double code(double kx, double ky, double th) {
	return (ky / fabs(kx)) * 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)) * th
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[(N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

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.64 N/A

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

   3. lower-pow.64 N/A

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

   4. lower-sin.64 36.2%

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

4. Applied rewrites 36.2%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 16.7%

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

7. Applied rewrites 16.7%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 13.7%

    \[\leadsto \frac{ky}{kx} \cdot \color{blue}{th} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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)))