Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 6.7s

Alternatives: 26

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

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: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

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

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

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 93.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

3. Applied rewrites 99.7%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 93.8

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

   7. lift-sqrt.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-hypot.f64 99.6

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

3. Applied rewrites 99.6%

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

Alternative 3: 80.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ t_3 := ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)\\ \mathbf{if}\;t\_2 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_2 \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.0065:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, t\_3\right)} \cdot t\_3\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;\frac{th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{1}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1))))
        (t_3 (* ky (+ 1.0 (* -0.16666666666666666 (pow ky 2.0))))))
   (if (<= t_2 -0.996)
     (* (/ (sin ky) (sqrt t_1)) (sin th))
     (if (<= t_2 -0.1)
       (* t_2 th)
       (if (<= t_2 0.0065)
         (* (/ (sin th) (hypot (sin kx) t_3)) t_3)
         (if (<= t_2 0.995)
           (/
            (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0))))
            (* (hypot (sin kx) (sin ky)) (/ 1.0 (sin ky))))
           (* (/ ky (hypot ky kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double t_3 = ky * (1.0 + (-0.16666666666666666 * pow(ky, 2.0)));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = t_2 * th;
	} else if (t_2 <= 0.0065) {
		tmp = (sin(th) / hypot(sin(kx), t_3)) * t_3;
	} else if (t_2 <= 0.995) {
		tmp = (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0)))) / (hypot(sin(kx), sin(ky)) * (1.0 / sin(ky)));
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double t_3 = ky * (1.0 + (-0.16666666666666666 * Math.pow(ky, 2.0)));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (Math.sin(ky) / Math.sqrt(t_1)) * Math.sin(th);
	} else if (t_2 <= -0.1) {
		tmp = t_2 * th;
	} else if (t_2 <= 0.0065) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), t_3)) * t_3;
	} else if (t_2 <= 0.995) {
		tmp = (th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0)))) / (Math.hypot(Math.sin(kx), Math.sin(ky)) * (1.0 / Math.sin(ky)));
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	t_3 = ky * (1.0 + (-0.16666666666666666 * math.pow(ky, 2.0)))
	tmp = 0
	if t_2 <= -0.996:
		tmp = (math.sin(ky) / math.sqrt(t_1)) * math.sin(th)
	elif t_2 <= -0.1:
		tmp = t_2 * th
	elif t_2 <= 0.0065:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), t_3)) * t_3
	elif t_2 <= 0.995:
		tmp = (th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0)))) / (math.hypot(math.sin(kx), math.sin(ky)) * (1.0 / math.sin(ky)))
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	t_3 = Float64(ky * Float64(1.0 + Float64(-0.16666666666666666 * (ky ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(t_2 * th);
	elseif (t_2 <= 0.0065)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), t_3)) * t_3);
	elseif (t_2 <= 0.995)
		tmp = Float64(Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))) / Float64(hypot(sin(kx), sin(ky)) * Float64(1.0 / sin(ky))));
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	t_3 = ky * (1.0 + (-0.16666666666666666 * (ky ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.996)
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	elseif (t_2 <= -0.1)
		tmp = t_2 * th;
	elseif (t_2 <= 0.0065)
		tmp = (sin(th) / hypot(sin(kx), t_3)) * t_3;
	elseif (t_2 <= 0.995)
		tmp = (th * (1.0 + (-0.16666666666666666 * (th ^ 2.0)))) / (hypot(sin(kx), sin(ky)) * (1.0 / sin(ky)));
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(ky * N[(1.0 + N[(-0.16666666666666666 * N[Power[ky, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(t$95$2 * th), $MachinePrecision], If[LessEqual[t$95$2, 0.0065], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$3 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 0.995], N[(N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 93.8%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.3

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

   4. Applied rewrites 40.3%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.8%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 48.6%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0064999999999999997

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      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. associate-/l* N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. pow2 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 53.1

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 54.6

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

   11. Applied rewrites 54.6%

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

   if 0.0064999999999999997 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.8%

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

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.8

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-hypot.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqr-neg-rev N/A

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

      7. sqr-neg-rev N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-hypot.f64 N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.4

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

   8. Applied rewrites 51.4%

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

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

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   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 34.7%

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

   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 47.6%

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

Alternative 4: 80.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ t_3 := t\_2 \cdot th\\ t_4 := ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)\\ \mathbf{if}\;t\_2 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.0065:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, t\_4\right)} \cdot t\_4\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1))))
        (t_3 (* t_2 th))
        (t_4 (* ky (+ 1.0 (* -0.16666666666666666 (pow ky 2.0))))))
   (if (<= t_2 -0.996)
     (* (/ (sin ky) (sqrt t_1)) (sin th))
     (if (<= t_2 -0.1)
       t_3
       (if (<= t_2 0.0065)
         (* (/ (sin th) (hypot (sin kx) t_4)) t_4)
         (if (<= t_2 0.995) t_3 (* (/ ky (hypot ky kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double t_3 = t_2 * th;
	double t_4 = ky * (1.0 + (-0.16666666666666666 * pow(ky, 2.0)));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = t_3;
	} else if (t_2 <= 0.0065) {
		tmp = (sin(th) / hypot(sin(kx), t_4)) * t_4;
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double t_3 = t_2 * th;
	double t_4 = ky * (1.0 + (-0.16666666666666666 * Math.pow(ky, 2.0)));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (Math.sin(ky) / Math.sqrt(t_1)) * Math.sin(th);
	} else if (t_2 <= -0.1) {
		tmp = t_3;
	} else if (t_2 <= 0.0065) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), t_4)) * t_4;
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	t_3 = t_2 * th
	t_4 = ky * (1.0 + (-0.16666666666666666 * math.pow(ky, 2.0)))
	tmp = 0
	if t_2 <= -0.996:
		tmp = (math.sin(ky) / math.sqrt(t_1)) * math.sin(th)
	elif t_2 <= -0.1:
		tmp = t_3
	elif t_2 <= 0.0065:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), t_4)) * t_4
	elif t_2 <= 0.995:
		tmp = t_3
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	t_3 = Float64(t_2 * th)
	t_4 = Float64(ky * Float64(1.0 + Float64(-0.16666666666666666 * (ky ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = t_3;
	elseif (t_2 <= 0.0065)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), t_4)) * t_4);
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	t_3 = t_2 * th;
	t_4 = ky * (1.0 + (-0.16666666666666666 * (ky ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.996)
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	elseif (t_2 <= -0.1)
		tmp = t_3;
	elseif (t_2 <= 0.0065)
		tmp = (sin(th) / hypot(sin(kx), t_4)) * t_4;
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * th), $MachinePrecision]}, Block[{t$95$4 = N[(ky * N[(1.0 + N[(-0.16666666666666666 * N[Power[ky, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], t$95$3, If[LessEqual[t$95$2, 0.0065], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$4 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$2, 0.995], t$95$3, N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 93.8%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.3

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

   4. Applied rewrites 40.3%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.0064999999999999997 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.8%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 48.6%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0064999999999999997

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      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. associate-/l* N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. pow2 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 53.1

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 54.6

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

   11. Applied rewrites 54.6%

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

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

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   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 34.7%

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

   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 47.6%

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

Alternative 5: 79.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;t\_3 \leq 0.0065:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (+ 1.0 (* -0.16666666666666666 (pow ky 2.0)))))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.996)
     (* (/ (sin ky) (sqrt t_2)) (sin th))
     (if (<= t_3 -0.1)
       (* (/ th (hypot (sin kx) (sin ky))) (sin ky))
       (if (<= t_3 0.0065)
         (* (/ (sin th) (hypot (sin kx) t_1)) t_1)
         (if (<= t_3 0.995)
           (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
           (* (/ ky (hypot ky kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * (1.0 + (-0.16666666666666666 * pow(ky, 2.0)));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	} else if (t_3 <= 0.0065) {
		tmp = (sin(th) / hypot(sin(kx), t_1)) * t_1;
	} else if (t_3 <= 0.995) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = ky * (1.0 + (-0.16666666666666666 * Math.pow(ky, 2.0)));
	double t_2 = Math.pow(Math.sin(ky), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.996) {
		tmp = (Math.sin(ky) / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (th / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	} else if (t_3 <= 0.0065) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), t_1)) * t_1;
	} else if (t_3 <= 0.995) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = ky * (1.0 + (-0.16666666666666666 * math.pow(ky, 2.0)))
	t_2 = math.pow(math.sin(ky), 2.0)
	t_3 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_2))
	tmp = 0
	if t_3 <= -0.996:
		tmp = (math.sin(ky) / math.sqrt(t_2)) * math.sin(th)
	elif t_3 <= -0.1:
		tmp = (th / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	elif t_3 <= 0.0065:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), t_1)) * t_1
	elif t_3 <= 0.995:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * Float64(1.0 + Float64(-0.16666666666666666 * (ky ^ 2.0))))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(th / hypot(sin(kx), sin(ky))) * sin(ky));
	elseif (t_3 <= 0.0065)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), t_1)) * t_1);
	elseif (t_3 <= 0.995)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = ky * (1.0 + (-0.16666666666666666 * (ky ^ 2.0)));
	t_2 = sin(ky) ^ 2.0;
	t_3 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_2));
	tmp = 0.0;
	if (t_3 <= -0.996)
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	elseif (t_3 <= -0.1)
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	elseif (t_3 <= 0.0065)
		tmp = (sin(th) / hypot(sin(kx), t_1)) * t_1;
	elseif (t_3 <= 0.995)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[(1.0 + N[(-0.16666666666666666 * N[Power[ky, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0065], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 93.8%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.3

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

   4. Applied rewrites 40.3%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      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. associate-/l* N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. pow2 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 51.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0064999999999999997

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      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. associate-/l* N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. pow2 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 53.1

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 54.6

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

   11. Applied rewrites 54.6%

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

   if 0.0064999999999999997 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.7%

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

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

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   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 34.7%

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

   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 47.6%

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

Alternative 6: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;t\_3 \leq 0.0065:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* ky (+ 1.0 (* -0.16666666666666666 (pow ky 2.0)))))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.996)
     (* (/ (sin ky) (sqrt t_2)) (sin th))
     (if (<= t_3 -0.1)
       (* (/ th (hypot (sin kx) (sin ky))) (sin ky))
       (if (<= t_3 0.0065)
         (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
         (if (<= t_3 0.995)
           (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
           (* (/ ky (hypot ky kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = ky * (1.0 + (-0.16666666666666666 * pow(ky, 2.0)));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	} else if (t_3 <= 0.0065) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else if (t_3 <= 0.995) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = ky * (1.0 + (-0.16666666666666666 * Math.pow(ky, 2.0)));
	double t_2 = Math.pow(Math.sin(ky), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.996) {
		tmp = (Math.sin(ky) / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (th / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	} else if (t_3 <= 0.0065) {
		tmp = (t_1 / Math.hypot(t_1, Math.sin(kx))) * Math.sin(th);
	} else if (t_3 <= 0.995) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = ky * (1.0 + (-0.16666666666666666 * math.pow(ky, 2.0)))
	t_2 = math.pow(math.sin(ky), 2.0)
	t_3 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_2))
	tmp = 0
	if t_3 <= -0.996:
		tmp = (math.sin(ky) / math.sqrt(t_2)) * math.sin(th)
	elif t_3 <= -0.1:
		tmp = (th / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	elif t_3 <= 0.0065:
		tmp = (t_1 / math.hypot(t_1, math.sin(kx))) * math.sin(th)
	elif t_3 <= 0.995:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(ky * Float64(1.0 + Float64(-0.16666666666666666 * (ky ^ 2.0))))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(th / hypot(sin(kx), sin(ky))) * sin(ky));
	elseif (t_3 <= 0.0065)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	elseif (t_3 <= 0.995)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = ky * (1.0 + (-0.16666666666666666 * (ky ^ 2.0)));
	t_2 = sin(ky) ^ 2.0;
	t_3 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_2));
	tmp = 0.0;
	if (t_3 <= -0.996)
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	elseif (t_3 <= -0.1)
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	elseif (t_3 <= 0.0065)
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	elseif (t_3 <= 0.995)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[(1.0 + N[(-0.16666666666666666 * N[Power[ky, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0065], 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$3, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 93.8%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.3

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

   4. Applied rewrites 40.3%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      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. associate-/l* N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. pow2 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 51.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0064999999999999997

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.9

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

   6. Applied rewrites 51.9%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.6

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

   9. Applied rewrites 55.6%

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

   if 0.0064999999999999997 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.7%

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

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

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   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 34.7%

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

   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 47.6%

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

Alternative 7: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.996)
     (* (/ (sin ky) (sqrt t_1)) (sin th))
     (if (<= t_2 -0.1)
       (* (/ th (hypot (sin kx) (sin ky))) (sin ky))
       (if (<= t_2 0.04)
         (/ (sin th) (/ (hypot (sin kx) ky) ky))
         (if (<= t_2 0.995)
           (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
           (* (/ ky (hypot ky kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	} else if (t_2 <= 0.04) {
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	} else if (t_2 <= 0.995) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (Math.sin(ky) / Math.sqrt(t_1)) * Math.sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (th / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	} else if (t_2 <= 0.04) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / ky);
	} else if (t_2 <= 0.995) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	tmp = 0
	if t_2 <= -0.996:
		tmp = (math.sin(ky) / math.sqrt(t_1)) * math.sin(th)
	elif t_2 <= -0.1:
		tmp = (th / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	elif t_2 <= 0.04:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / ky)
	elif t_2 <= 0.995:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(th / hypot(sin(kx), sin(ky))) * sin(ky));
	elseif (t_2 <= 0.04)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / ky));
	elseif (t_2 <= 0.995)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	tmp = 0.0;
	if (t_2 <= -0.996)
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	elseif (t_2 <= -0.1)
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	elseif (t_2 <= 0.04)
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	elseif (t_2 <= 0.995)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.04], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 93.8%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.3

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

   4. Applied rewrites 40.3%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      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. associate-/l* N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. pow2 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 51.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0400000000000000008

   1. Initial program 93.8%

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

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.8

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 66.2%

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

   if 0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.7%

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

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

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   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 34.7%

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

   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 47.6%

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

Alternative 8: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.996:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_1}}\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.996)
     (/ (* (sin ky) (sin th)) (sqrt t_1))
     (if (<= t_2 -0.1)
       (* (/ th (hypot (sin kx) (sin ky))) (sin ky))
       (if (<= t_2 0.04)
         (/ (sin th) (/ (hypot (sin kx) ky) ky))
         (if (<= t_2 0.995)
           (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
           (* (/ ky (hypot ky kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_1);
	} else if (t_2 <= -0.1) {
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	} else if (t_2 <= 0.04) {
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	} else if (t_2 <= 0.995) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sqrt(t_1);
	} else if (t_2 <= -0.1) {
		tmp = (th / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	} else if (t_2 <= 0.04) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / ky);
	} else if (t_2 <= 0.995) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	tmp = 0
	if t_2 <= -0.996:
		tmp = (math.sin(ky) * math.sin(th)) / math.sqrt(t_1)
	elif t_2 <= -0.1:
		tmp = (th / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	elif t_2 <= 0.04:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / ky)
	elif t_2 <= 0.995:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.996)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_1));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(th / hypot(sin(kx), sin(ky))) * sin(ky));
	elseif (t_2 <= 0.04)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / ky));
	elseif (t_2 <= 0.995)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	tmp = 0.0;
	if (t_2 <= -0.996)
		tmp = (sin(ky) * sin(th)) / sqrt(t_1);
	elseif (t_2 <= -0.1)
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	elseif (t_2 <= 0.04)
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	elseif (t_2 <= 0.995)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.04], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 93.8%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.0

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

   4. Applied rewrites 41.0%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      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. associate-/l* N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. pow2 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 51.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0400000000000000008

   1. Initial program 93.8%

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

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.8

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 66.2%

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

   if 0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.7%

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

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

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   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 34.7%

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

   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 47.6%

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

Alternative 9: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.9995)
     (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
     (if (<= t_1 -0.1)
       (* (/ th (hypot (sin kx) (sin ky))) (sin ky))
       (if (<= t_1 0.04)
         (/ (sin th) (/ (hypot (sin kx) ky) ky))
         (if (<= t_1 0.995)
           (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
           (* (/ ky (hypot ky kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.9995) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	} else if (t_1 <= 0.04) {
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	} else if (t_1 <= 0.995) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.9995) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	} else if (t_1 <= -0.1) {
		tmp = (th / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	} else if (t_1 <= 0.04) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / ky);
	} else if (t_1 <= 0.995) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.9995:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	elif t_1 <= -0.1:
		tmp = (th / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	elif t_1 <= 0.04:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / ky)
	elif t_1 <= 0.995:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.9995)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = Float64(Float64(th / hypot(sin(kx), sin(ky))) * sin(ky));
	elseif (t_1 <= 0.04)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / ky));
	elseif (t_1 <= 0.995)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.9995)
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	elseif (t_1 <= -0.1)
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	elseif (t_1 <= 0.04)
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	elseif (t_1 <= 0.995)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.04], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99950000000000006

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   if -0.99950000000000006 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      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. associate-/l* N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. pow2 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 51.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0400000000000000008

   1. Initial program 93.8%

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

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.8

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 66.2%

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

   if 0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.7%

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

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

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   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 34.7%

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

   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 47.6%

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

Alternative 10: 77.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{if}\;t\_1 \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (* (/ th (hypot (sin kx) (sin ky))) (sin ky))))
   (if (<= t_1 -0.9995)
     (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
     (if (<= t_1 -0.1)
       t_2
       (if (<= t_1 0.04)
         (/ (sin th) (/ (hypot (sin kx) ky) ky))
         (if (<= t_1 0.995) t_2 (* (/ ky (hypot ky kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	double tmp;
	if (t_1 <= -0.9995) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = t_2;
	} else if (t_1 <= 0.04) {
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	} else if (t_1 <= 0.995) {
		tmp = t_2;
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_2 = (th / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	double tmp;
	if (t_1 <= -0.9995) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	} else if (t_1 <= -0.1) {
		tmp = t_2;
	} else if (t_1 <= 0.04) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / ky);
	} else if (t_1 <= 0.995) {
		tmp = t_2;
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_2 = (th / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	tmp = 0
	if t_1 <= -0.9995:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	elif t_1 <= -0.1:
		tmp = t_2
	elif t_1 <= 0.04:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / ky)
	elif t_1 <= 0.995:
		tmp = t_2
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(th / hypot(sin(kx), sin(ky))) * sin(ky))
	tmp = 0.0
	if (t_1 <= -0.9995)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = t_2;
	elseif (t_1 <= 0.04)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / ky));
	elseif (t_1 <= 0.995)
		tmp = t_2;
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_2 = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	tmp = 0.0;
	if (t_1 <= -0.9995)
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	elseif (t_1 <= -0.1)
		tmp = t_2;
	elseif (t_1 <= 0.04)
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	elseif (t_1 <= 0.995)
		tmp = t_2;
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], t$95$2, If[LessEqual[t$95$1, 0.04], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], t$95$2, N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99950000000000006

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   if -0.99950000000000006 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      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. associate-/l* N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. pow2 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 51.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0400000000000000008

   1. Initial program 93.8%

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

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.8

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 66.2%

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

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

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   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 34.7%

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

   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 47.6%

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

Alternative 11: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{if}\;t\_1 \leq -0.9995:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.04:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (* (/ th (hypot (sin kx) (sin ky))) (sin ky))))
   (if (<= t_1 -0.9995)
     (/ (* (sin th) (sin ky)) (hypot kx (sin ky)))
     (if (<= t_1 -0.1)
       t_2
       (if (<= t_1 0.04)
         (/ (sin th) (/ (hypot (sin kx) ky) ky))
         (if (<= t_1 0.995) t_2 (* (/ ky (hypot ky kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	double tmp;
	if (t_1 <= -0.9995) {
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	} else if (t_1 <= -0.1) {
		tmp = t_2;
	} else if (t_1 <= 0.04) {
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	} else if (t_1 <= 0.995) {
		tmp = t_2;
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_2 = (th / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	double tmp;
	if (t_1 <= -0.9995) {
		tmp = (Math.sin(th) * Math.sin(ky)) / Math.hypot(kx, Math.sin(ky));
	} else if (t_1 <= -0.1) {
		tmp = t_2;
	} else if (t_1 <= 0.04) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / ky);
	} else if (t_1 <= 0.995) {
		tmp = t_2;
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_2 = (th / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	tmp = 0
	if t_1 <= -0.9995:
		tmp = (math.sin(th) * math.sin(ky)) / math.hypot(kx, math.sin(ky))
	elif t_1 <= -0.1:
		tmp = t_2
	elif t_1 <= 0.04:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / ky)
	elif t_1 <= 0.995:
		tmp = t_2
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(th / hypot(sin(kx), sin(ky))) * sin(ky))
	tmp = 0.0
	if (t_1 <= -0.9995)
		tmp = Float64(Float64(sin(th) * sin(ky)) / hypot(kx, sin(ky)));
	elseif (t_1 <= -0.1)
		tmp = t_2;
	elseif (t_1 <= 0.04)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / ky));
	elseif (t_1 <= 0.995)
		tmp = t_2;
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_2 = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	tmp = 0.0;
	if (t_1 <= -0.9995)
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	elseif (t_1 <= -0.1)
		tmp = t_2;
	elseif (t_1 <= 0.04)
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	elseif (t_1 <= 0.995)
		tmp = t_2;
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9995], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], t$95$2, If[LessEqual[t$95$1, 0.04], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], t$95$2, N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99950000000000006

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.1

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 54.2

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

   6. Applied rewrites 54.2%

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

   if -0.99950000000000006 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.0400000000000000008 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      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. associate-/l* N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. pow2 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 51.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0400000000000000008

   1. Initial program 93.8%

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

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.8

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 66.2%

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

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

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.1

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

   6. Applied rewrites 58.1%

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

   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 34.7%

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

   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 47.6%

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

Alternative 12: 69.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= -0.1:
		tmp = (math.sin(th) * math.sin(ky)) / math.hypot(kx, math.sin(ky))
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.1)
		tmp = Float64(Float64(sin(th) * sin(ky)) / hypot(kx, sin(ky)));
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.1)
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	else
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[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], -0.1], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.8%

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

   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 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.1

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 54.2

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

   6. Applied rewrites 54.2%

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

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

   1. Initial program 93.8%

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

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.8

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 66.2%

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

Alternative 13: 66.2% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.1)
   (* (/ (sin ky) (sqrt (pow (sin ky) 2.0))) th)
   (/ (sin th) (/ (hypot (sin kx) ky) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.1) {
		tmp = (sin(ky) / sqrt(pow(sin(ky), 2.0))) * th;
	} else {
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	}
	return tmp;
}

Java

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

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.1:
		tmp = (math.sin(ky) / math.sqrt(math.pow(math.sin(ky), 2.0))) * th
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt((sin(ky) ^ 2.0))) * th);
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.1)
		tmp = (sin(ky) / sqrt((sin(ky) ^ 2.0))) * th;
	else
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.10000000000000001

   1. Initial program 93.8%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.8

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

   7. Applied rewrites 16.8%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 13.9%

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

   11. Taylor expanded in kx around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin ky}^{2}}}} \cdot th \]

       2. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot th \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot th \]

       4. lower-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot th \]

       5. lower-sin.f64 21.8

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot th \]

   13. Applied rewrites 21.8%

       \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin ky}^{2}}}} \cdot th \]

   if -0.10000000000000001 < (sin.f64 ky)

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. div-flip N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 93.8

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}}{\sin ky}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{{\sin ky}^{2}}}}{\sin ky}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

      14. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   4. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \color{blue}{ky}\right)}{\sin ky}} \]
   5. Step-by-step derivation

   6. Applied rewrites 53.4%

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \color{blue}{ky}\right)}{\sin ky}} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\color{blue}{ky}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 66.2%

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\color{blue}{ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 66.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.1:\\ \;\;\;\;\left(\frac{1}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin ky\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.1)
   (* (* (/ 1.0 (hypot kx (sin ky))) (sin ky)) th)
   (/ (sin th) (/ (hypot (sin kx) ky) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.1) {
		tmp = ((1.0 / hypot(kx, sin(ky))) * sin(ky)) * th;
	} else {
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.1) {
		tmp = ((1.0 / Math.hypot(kx, Math.sin(ky))) * Math.sin(ky)) * th;
	} else {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.1:
		tmp = ((1.0 / math.hypot(kx, math.sin(ky))) * math.sin(ky)) * th
	else:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), ky) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.1)
		tmp = Float64(Float64(Float64(1.0 / hypot(kx, sin(ky))) * sin(ky)) * th);
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), ky) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.1)
		tmp = ((1.0 / hypot(kx, sin(ky))) * sin(ky)) * th;
	else
		tmp = sin(th) / (hypot(sin(kx), ky) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.1], N[(N[(N[(1.0 / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.10000000000000001

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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 52.2%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{2}{2}}}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{2}{2}}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      7. metadata-eval 52.1

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      9. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      10. lift-pow.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      12. lift-pow.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin ky\right) \cdot \sin th \]

      13. pow2 N/A

          \[\leadsto \left(\frac{1}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin ky\right) \cdot \sin th \]

      14. lower-hypot.f64 58.0

          \[\leadsto \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin ky\right) \cdot \sin th \]

   6. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin ky\right)} \cdot \sin th \]

   7. Taylor expanded in th around 0

      \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin ky\right) \cdot \color{blue}{th} \]
   8. Step-by-step derivation

   9. Applied rewrites 34.6%

      \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin ky\right) \cdot \color{blue}{th} \]

   if -0.10000000000000001 < (sin.f64 ky)

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. div-flip N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 93.8

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}}{\sin ky}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{{\sin ky}^{2}}}}{\sin ky}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

      14. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   4. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \color{blue}{ky}\right)}{\sin ky}} \]
   5. Step-by-step derivation

   6. Applied rewrites 53.4%

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \color{blue}{ky}\right)}{\sin ky}} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\color{blue}{ky}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 66.2%

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\color{blue}{ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (/ (sin th) (/ (hypot (sin kx) ky) ky)))

C

double code(double kx, double ky, double th) {
	return sin(th) / (hypot(sin(kx), ky) / ky);
}

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th) / (Math.hypot(Math.sin(kx), ky) / ky);
}

Python

def code(kx, ky, th):
	return math.sin(th) / (math.hypot(math.sin(kx), ky) / ky)

Julia

function code(kx, ky, th)
	return Float64(sin(th) / Float64(hypot(sin(kx), ky) / ky))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th) / (hypot(sin(kx), ky) / ky);
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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. *-commutative N/A

      \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. lift-/.f64 N/A

      \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   4. div-flip N/A

      \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   5. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   7. lower-/.f64 93.8

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}}{\sin ky}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{{\sin ky}^{2}}}}{\sin ky}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

   14. lower-hypot.f64 99.6

       \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

4. Taylor expanded in ky around 0

   \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \color{blue}{ky}\right)}{\sin ky}} \]
5. Step-by-step derivation

6. Applied rewrites 53.4%

   \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \color{blue}{ky}\right)}{\sin ky}} \]

7. Taylor expanded in ky around 0

   \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\color{blue}{ky}}} \]
8. Step-by-step derivation

9. Applied rewrites 66.2%

   \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{\color{blue}{ky}}} \]
10. Add Preprocessing

Alternative 16: 65.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th \end{array} \]

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 93.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   3. +-commutative N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

   5. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

   7. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

   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 52.2%

   \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

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 66.2%

   \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
10. Add Preprocessing

Alternative 17: 63.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.01:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 0.01)
   (* (/ ky (hypot ky kx)) (sin th))
   (* (/ ky (sqrt (- 0.5 (* 0.5 (cos (+ kx kx)))))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 0.01) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = (ky / sqrt((0.5 - (0.5 * cos((kx + kx)))))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(kx), 2.0) <= 0.01) {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	} else {
		tmp = (ky / Math.sqrt((0.5 - (0.5 * Math.cos((kx + kx)))))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(kx), 2.0) <= 0.01:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	else:
		tmp = (ky / math.sqrt((0.5 - (0.5 * math.cos((kx + kx)))))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 0.01)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx)))))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(kx) ^ 2.0) <= 0.01)
		tmp = (ky / hypot(ky, kx)) * sin(th);
	else
		tmp = (ky / sqrt((0.5 - (0.5 * cos((kx + kx)))))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 0.01], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.0100000000000000002

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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 52.2%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {kx}^{2}}} \cdot \sin th \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \sin th \]

      8. lower-hypot.f64 58.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   6. Applied rewrites 58.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   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 34.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   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 47.6%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]

   if 0.0100000000000000002 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      4. lower-sin.f64 36.7

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

      10. lower-+.f64 27.3

          \[\leadsto \frac{ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

   6. Applied rewrites 27.3%

      \[\leadsto \frac{ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 58.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\left(\frac{1}{\left|\sin kx\right|} \cdot \sin ky\right) \cdot th\\ \mathbf{elif}\;\sin kx \leq 0.115:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin kx}{ky}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (* (* (/ 1.0 (fabs (sin kx))) (sin ky)) th)
   (if (<= (sin kx) 0.115)
     (* (/ ky (hypot ky kx)) (sin th))
     (* (/ 1.0 (/ (sin kx) ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = ((1.0 / fabs(sin(kx))) * sin(ky)) * th;
	} else if (sin(kx) <= 0.115) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = (1.0 / (sin(kx) / ky)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.15) {
		tmp = ((1.0 / Math.abs(Math.sin(kx))) * Math.sin(ky)) * th;
	} else if (Math.sin(kx) <= 0.115) {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	} else {
		tmp = (1.0 / (Math.sin(kx) / ky)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.15:
		tmp = ((1.0 / math.fabs(math.sin(kx))) * math.sin(ky)) * th
	elif math.sin(kx) <= 0.115:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	else:
		tmp = (1.0 / (math.sin(kx) / ky)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.15)
		tmp = Float64(Float64(Float64(1.0 / abs(sin(kx))) * sin(ky)) * th);
	elseif (sin(kx) <= 0.115)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(Float64(1.0 / Float64(sin(kx) / ky)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.15)
		tmp = ((1.0 / abs(sin(kx))) * sin(ky)) * th;
	elseif (sin(kx) <= 0.115)
		tmp = (ky / hypot(ky, kx)) * sin(th);
	else
		tmp = (1.0 / (sin(kx) / ky)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.15], N[(N[(N[(1.0 / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.115], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-sin.f64 41.7

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 41.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\sqrt{{\sin kx}^{2}}}\right)} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{{\sin kx}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{{\sin kx}^{2}}} \cdot \sin ky\right)} \cdot \sin th \]

   6. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sin kx} \cdot \sin ky\right)} \cdot \sin th \]

   7. Taylor expanded in th around 0

      \[\leadsto \left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \color{blue}{th} \]
   8. Step-by-step derivation

   9. Applied rewrites 17.9%

      \[\leadsto \left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \color{blue}{th} \]
   10. Step-by-step derivation

       1. unpow1 N/A

          \[\leadsto \left(\frac{1}{{\sin kx}^{\color{blue}{1}}} \cdot \sin ky\right) \cdot th \]

       2. sqr-pow N/A

          \[\leadsto \left(\frac{1}{{\sin kx}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\sin kx}^{\left(\frac{1}{2}\right)}}} \cdot \sin ky\right) \cdot th \]

       3. metadata-eval N/A

          \[\leadsto \left(\frac{1}{{\sin kx}^{\frac{1}{2}} \cdot {\sin kx}^{\left(\frac{1}{2}\right)}} \cdot \sin ky\right) \cdot th \]

       4. metadata-eval N/A

          \[\leadsto \left(\frac{1}{{\sin kx}^{\frac{1}{2}} \cdot {\sin kx}^{\frac{1}{2}}} \cdot \sin ky\right) \cdot th \]

       5. unpow-prod-down N/A

          \[\leadsto \left(\frac{1}{{\left(\sin kx \cdot \sin kx\right)}^{\color{blue}{\frac{1}{2}}}} \cdot \sin ky\right) \cdot th \]

       6. pow2 N/A

          \[\leadsto \left(\frac{1}{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}} \cdot \sin ky\right) \cdot th \]

       7. lift-sin.f64 N/A

          \[\leadsto \left(\frac{1}{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}} \cdot \sin ky\right) \cdot th \]

       8. pow1/2 N/A

          \[\leadsto \left(\frac{1}{\sqrt{{\sin kx}^{2}}} \cdot \sin ky\right) \cdot th \]

       9. lift-sin.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{{\sin kx}^{2}}} \cdot \sin ky\right) \cdot th \]

       10. pow2 N/A

           \[\leadsto \left(\frac{1}{\sqrt{\sin kx \cdot \sin kx}} \cdot \sin ky\right) \cdot th \]

       11. rem-sqrt-square-rev N/A

           \[\leadsto \left(\frac{1}{\left|\sin kx\right|} \cdot \sin ky\right) \cdot th \]

       12. lower-fabs.f64 24.2

           \[\leadsto \left(\frac{1}{\left|\sin kx\right|} \cdot \sin ky\right) \cdot th \]

   11. Applied rewrites 24.2%

       \[\leadsto \left(\frac{1}{\left|\sin kx\right|} \cdot \sin ky\right) \cdot th \]

   if -0.149999999999999994 < (sin.f64 kx) < 0.115000000000000005

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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 52.2%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {kx}^{2}}} \cdot \sin th \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \sin th \]

      8. lower-hypot.f64 58.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   6. Applied rewrites 58.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   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 34.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   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 47.6%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]

   if 0.115000000000000005 < (sin.f64 kx)

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      4. lower-sin.f64 36.7

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2}}}{ky}}} \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{2}{2}}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{ky}} \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2}{2}}{\color{blue}{\frac{\sqrt{{\sin kx}^{2}}}{ky}}} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{ky}} \cdot \sin th \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{{\sin kx}^{2}}}{ky}} \cdot \sin th \]

      7. pow1/2 N/A

         \[\leadsto \frac{1}{\frac{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      9. pow2 N/A

         \[\leadsto \frac{1}{\frac{{\left(\sin kx \cdot \sin kx\right)}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      10. unpow-prod-down N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\frac{1}{2}} \cdot {\sin kx}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\left(\frac{1}{2}\right)} \cdot {\sin kx}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\sin kx}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      13. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\sin kx}^{\left(\frac{1}{2}\right)}}{ky}} \cdot \sin th \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)}}{ky}} \cdot \sin th \]

      15. sqr-pow N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\left(\frac{2}{2}\right)}}{ky}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{1}}{ky}} \cdot \sin th \]

      17. unpow1 N/A

          \[\leadsto \frac{1}{\frac{\sin kx}{ky}} \cdot \sin th \]

      18. lower-/.f64 25.6

          \[\leadsto \frac{1}{\frac{\sin kx}{\color{blue}{ky}}} \cdot \sin th \]

   6. Applied rewrites 25.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin kx}{ky}}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 57.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot th\\ \mathbf{elif}\;\sin kx \leq 0.115:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin kx}{ky}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (* (/ ky (sqrt (pow (sin kx) 2.0))) th)
   (if (<= (sin kx) 0.115)
     (* (/ ky (hypot ky kx)) (sin th))
     (* (/ 1.0 (/ (sin kx) ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = (ky / sqrt(pow(sin(kx), 2.0))) * th;
	} else if (sin(kx) <= 0.115) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = (1.0 / (sin(kx) / ky)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.15) {
		tmp = (ky / Math.sqrt(Math.pow(Math.sin(kx), 2.0))) * th;
	} else if (Math.sin(kx) <= 0.115) {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	} else {
		tmp = (1.0 / (Math.sin(kx) / ky)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.15:
		tmp = (ky / math.sqrt(math.pow(math.sin(kx), 2.0))) * th
	elif math.sin(kx) <= 0.115:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	else:
		tmp = (1.0 / (math.sin(kx) / ky)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.15)
		tmp = Float64(Float64(ky / sqrt((sin(kx) ^ 2.0))) * th);
	elseif (sin(kx) <= 0.115)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(Float64(1.0 / Float64(sin(kx) / ky)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.15)
		tmp = (ky / sqrt((sin(kx) ^ 2.0))) * th;
	elseif (sin(kx) <= 0.115)
		tmp = (ky / hypot(ky, kx)) * sin(th);
	else
		tmp = (1.0 / (sin(kx) / ky)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.15], N[(N[(ky / N[Sqrt[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.115], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      4. lower-sin.f64 36.7

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

   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 19.9%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{th} \]

   if -0.149999999999999994 < (sin.f64 kx) < 0.115000000000000005

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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 52.2%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {kx}^{2}}} \cdot \sin th \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \sin th \]

      8. lower-hypot.f64 58.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   6. Applied rewrites 58.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   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 34.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   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 47.6%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]

   if 0.115000000000000005 < (sin.f64 kx)

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      4. lower-sin.f64 36.7

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. div-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2}}}{ky}}} \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{2}{2}}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{ky}} \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2}{2}}{\color{blue}{\frac{\sqrt{{\sin kx}^{2}}}{ky}}} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2}}}}{ky}} \cdot \sin th \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{{\sin kx}^{2}}}{ky}} \cdot \sin th \]

      7. pow1/2 N/A

         \[\leadsto \frac{1}{\frac{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      9. pow2 N/A

         \[\leadsto \frac{1}{\frac{{\left(\sin kx \cdot \sin kx\right)}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      10. unpow-prod-down N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\frac{1}{2}} \cdot {\sin kx}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\left(\frac{1}{2}\right)} \cdot {\sin kx}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\sin kx}^{\frac{1}{2}}}{ky}} \cdot \sin th \]

      13. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\sin kx}^{\left(\frac{1}{2}\right)}}{ky}} \cdot \sin th \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)}}{ky}} \cdot \sin th \]

      15. sqr-pow N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{\left(\frac{2}{2}\right)}}{ky}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{{\sin kx}^{1}}{ky}} \cdot \sin th \]

      17. unpow1 N/A

          \[\leadsto \frac{1}{\frac{\sin kx}{ky}} \cdot \sin th \]

      18. lower-/.f64 25.6

          \[\leadsto \frac{1}{\frac{\sin kx}{\color{blue}{ky}}} \cdot \sin th \]

   6. Applied rewrites 25.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin kx}{ky}}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 57.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.15:\\ \;\;\;\;\frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot th\\ \mathbf{elif}\;\sin kx \leq 0.115:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.15)
   (* (/ ky (sqrt (pow (sin kx) 2.0))) th)
   (if (<= (sin kx) 0.115)
     (* (/ ky (hypot ky kx)) (sin th))
     (* (sin th) (/ ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.15) {
		tmp = (ky / sqrt(pow(sin(kx), 2.0))) * th;
	} else if (sin(kx) <= 0.115) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.15) {
		tmp = (ky / Math.sqrt(Math.pow(Math.sin(kx), 2.0))) * th;
	} else if (Math.sin(kx) <= 0.115) {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.15:
		tmp = (ky / math.sqrt(math.pow(math.sin(kx), 2.0))) * th
	elif math.sin(kx) <= 0.115:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.15)
		tmp = Float64(Float64(ky / sqrt((sin(kx) ^ 2.0))) * th);
	elseif (sin(kx) <= 0.115)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.15)
		tmp = (ky / sqrt((sin(kx) ^ 2.0))) * th;
	elseif (sin(kx) <= 0.115)
		tmp = (ky / hypot(ky, kx)) * sin(th);
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.15], N[(N[(ky / N[Sqrt[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.115], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.149999999999999994

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      4. lower-sin.f64 36.7

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

   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 19.9%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{th} \]

   if -0.149999999999999994 < (sin.f64 kx) < 0.115000000000000005

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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 52.2%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {kx}^{2}}} \cdot \sin th \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \sin th \]

      8. lower-hypot.f64 58.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   6. Applied rewrites 58.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   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 34.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   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 47.6%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]

   if 0.115000000000000005 < (sin.f64 kx)

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      4. lower-sin.f64 36.7

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   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 36.7

         \[\leadsto \color{blue}{\sin th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \sin th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}} \]

      5. pow1/2 N/A

         \[\leadsto \sin th \cdot \frac{ky}{{\left({\sin kx}^{2}\right)}^{\color{blue}{\frac{1}{2}}}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \sin th \cdot \frac{ky}{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}} \]

      7. pow2 N/A

         \[\leadsto \sin th \cdot \frac{ky}{{\left(\sin kx \cdot \sin kx\right)}^{\frac{1}{2}}} \]

      8. unpow-prod-down N/A

         \[\leadsto \sin th \cdot \frac{ky}{{\sin kx}^{\frac{1}{2}} \cdot \color{blue}{{\sin kx}^{\frac{1}{2}}}} \]

      9. metadata-eval N/A

         \[\leadsto \sin th \cdot \frac{ky}{{\sin kx}^{\left(\frac{1}{2}\right)} \cdot {\sin kx}^{\frac{1}{2}}} \]

      10. metadata-eval N/A

          \[\leadsto \sin th \cdot \frac{ky}{{\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\sin kx}^{\frac{1}{2}}} \]

      11. metadata-eval N/A

          \[\leadsto \sin th \cdot \frac{ky}{{\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\sin kx}^{\left(\frac{1}{\color{blue}{2}}\right)}} \]

      12. metadata-eval N/A

          \[\leadsto \sin th \cdot \frac{ky}{{\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {\sin kx}^{\left(\frac{\frac{2}{2}}{2}\right)}} \]

      13. sqr-pow N/A

          \[\leadsto \sin th \cdot \frac{ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      14. metadata-eval N/A

          \[\leadsto \sin th \cdot \frac{ky}{{\sin kx}^{1}} \]

      15. unpow1 25.6

          \[\leadsto \sin th \cdot \frac{ky}{\sin kx} \]

   6. Applied rewrites 25.6%

      \[\leadsto \color{blue}{\sin th \cdot \frac{ky}{\sin kx}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 49.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8 \cdot 10^{+39}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8e+39)
   (* (/ ky (hypot ky kx)) (sin th))
   (* (/ ky (sqrt (pow (sin kx) 2.0))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8e+39) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = (ky / sqrt(pow(sin(kx), 2.0))) * th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8e+39) {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	} else {
		tmp = (ky / Math.sqrt(Math.pow(Math.sin(kx), 2.0))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 8e+39:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	else:
		tmp = (ky / math.sqrt(math.pow(math.sin(kx), 2.0))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8e+39)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt((sin(kx) ^ 2.0))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 8e+39)
		tmp = (ky / hypot(ky, kx)) * sin(th);
	else
		tmp = (ky / sqrt((sin(kx) ^ 2.0))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8e+39], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 7.99999999999999952e39

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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 52.2%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {kx}^{2}}} \cdot \sin th \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \sin th \]

      8. lower-hypot.f64 58.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   6. Applied rewrites 58.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   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 34.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   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 47.6%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]

   if 7.99999999999999952e39 < kx

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      4. lower-sin.f64 36.7

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

   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 19.9%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 49.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8 \cdot 10^{+39}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{-\left(0.5 \cdot \cos \left(kx + kx\right) - 0.5\right)}} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8e+39)
   (* (/ ky (hypot ky kx)) (sin th))
   (* (/ ky (sqrt (- (- (* 0.5 (cos (+ kx kx))) 0.5)))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8e+39) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = (ky / sqrt(-((0.5 * cos((kx + kx))) - 0.5))) * th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8e+39) {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	} else {
		tmp = (ky / Math.sqrt(-((0.5 * Math.cos((kx + kx))) - 0.5))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 8e+39:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	else:
		tmp = (ky / math.sqrt(-((0.5 * math.cos((kx + kx))) - 0.5))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8e+39)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt(Float64(-Float64(Float64(0.5 * cos(Float64(kx + kx))) - 0.5)))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 8e+39)
		tmp = (ky / hypot(ky, kx)) * sin(th);
	else
		tmp = (ky / sqrt(-((0.5 * cos((kx + kx))) - 0.5))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8e+39], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[(-N[(N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 7.99999999999999952e39

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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 52.2%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {kx}^{2}}} \cdot \sin th \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \sin th \]

      8. lower-hypot.f64 58.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   6. Applied rewrites 58.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \sin th \]

   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 34.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   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 47.6%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]

   if 7.99999999999999952e39 < kx

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      4. lower-sin.f64 36.7

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\mathsf{neg}\left(\sin kx\right)\right) \cdot \left(\mathsf{neg}\left(\sin kx\right)\right)}} \cdot \sin th \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{ky}{\sqrt{\mathsf{neg}\left(\sin kx \cdot \left(\mathsf{neg}\left(\sin kx\right)\right)\right)}} \cdot \sin th \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \frac{ky}{\sqrt{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin kx \cdot \sin kx\right)\right)\right)}} \cdot \sin th \]

      6. pow2 N/A

         \[\leadsto \frac{ky}{\sqrt{\mathsf{neg}\left(\left(\mathsf{neg}\left({\sin kx}^{2}\right)\right)\right)}} \cdot \sin th \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\mathsf{neg}\left(\left(\mathsf{neg}\left({\sin kx}^{2}\right)\right)\right)}} \cdot \sin th \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left({\sin kx}^{2}\right)\right)}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left({\sin kx}^{2}\right)\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left(\sin kx \cdot \sin kx\right)\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left(\sin kx \cdot \sin kx\right)\right)}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left(\sin kx \cdot \sin kx\right)\right)}} \cdot \sin th \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)\right)}} \cdot \sin th \]

      14. sub-negate N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(2 \cdot kx\right) - \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower--.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(2 \cdot kx\right) - \frac{1}{2}\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(kx + kx\right) - \frac{1}{2}\right)}} \cdot \sin th \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(kx + kx\right) - \frac{1}{2}\right)}} \cdot \sin th \]

      18. lower-cos.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(kx + kx\right) - \frac{1}{2}\right)}} \cdot \sin th \]

      19. lower-+.f64 26.9

          \[\leadsto \frac{ky}{\sqrt{-\left(0.5 \cdot \cos \left(kx + kx\right) - 0.5\right)}} \cdot \sin th \]

   6. Applied rewrites 26.9%

      \[\leadsto \frac{ky}{\sqrt{-\left(0.5 \cdot \cos \left(kx + kx\right) - 0.5\right)}} \cdot \sin th \]

   7. Taylor expanded in th around 0

      \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(kx + kx\right) - \frac{1}{2}\right)}} \cdot \color{blue}{th} \]
   8. Step-by-step derivation

   9. Applied rewrites 14.8%

      \[\leadsto \frac{ky}{\sqrt{-\left(0.5 \cdot \cos \left(kx + kx\right) - 0.5\right)}} \cdot \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 19.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8 \cdot 10^{+39}:\\ \;\;\;\;\left(ky \cdot \frac{1}{kx}\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{-\left(0.5 \cdot \cos \left(kx + kx\right) - 0.5\right)}} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8e+39)
   (* (* ky (/ 1.0 kx)) (sin th))
   (* (/ ky (sqrt (- (- (* 0.5 (cos (+ kx kx))) 0.5)))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8e+39) {
		tmp = (ky * (1.0 / kx)) * sin(th);
	} else {
		tmp = (ky / sqrt(-((0.5 * cos((kx + kx))) - 0.5))) * 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 (kx <= 8d+39) then
        tmp = (ky * (1.0d0 / kx)) * sin(th)
    else
        tmp = (ky / sqrt(-((0.5d0 * cos((kx + kx))) - 0.5d0))) * th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8e+39) {
		tmp = (ky * (1.0 / kx)) * Math.sin(th);
	} else {
		tmp = (ky / Math.sqrt(-((0.5 * Math.cos((kx + kx))) - 0.5))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 8e+39:
		tmp = (ky * (1.0 / kx)) * math.sin(th)
	else:
		tmp = (ky / math.sqrt(-((0.5 * math.cos((kx + kx))) - 0.5))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8e+39)
		tmp = Float64(Float64(ky * Float64(1.0 / kx)) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt(Float64(-Float64(Float64(0.5 * cos(Float64(kx + kx))) - 0.5)))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 8e+39)
		tmp = (ky * (1.0 / kx)) * sin(th);
	else
		tmp = (ky / sqrt(-((0.5 * cos((kx + kx))) - 0.5))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8e+39], N[(N[(ky * N[(1.0 / kx), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[(-N[(N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 7.99999999999999952e39

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      4. lower-sin.f64 36.7

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-/.f64 16.8

         \[\leadsto \frac{ky}{kx} \cdot \sin th \]

   7. Applied rewrites 16.8%

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{ky}{kx} \cdot \sin th \]

      2. mult-flip N/A

         \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{kx}}\right) \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{kx}}\right) \cdot \sin th \]

      4. lower-/.f64 16.8

         \[\leadsto \left(ky \cdot \frac{1}{kx}\right) \cdot \sin th \]

   9. Applied rewrites 16.8%

      \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{kx}}\right) \cdot \sin th \]

   if 7.99999999999999952e39 < kx

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      4. lower-sin.f64 36.7

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\mathsf{neg}\left(\sin kx\right)\right) \cdot \left(\mathsf{neg}\left(\sin kx\right)\right)}} \cdot \sin th \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \frac{ky}{\sqrt{\mathsf{neg}\left(\sin kx \cdot \left(\mathsf{neg}\left(\sin kx\right)\right)\right)}} \cdot \sin th \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \frac{ky}{\sqrt{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin kx \cdot \sin kx\right)\right)\right)}} \cdot \sin th \]

      6. pow2 N/A

         \[\leadsto \frac{ky}{\sqrt{\mathsf{neg}\left(\left(\mathsf{neg}\left({\sin kx}^{2}\right)\right)\right)}} \cdot \sin th \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\mathsf{neg}\left(\left(\mathsf{neg}\left({\sin kx}^{2}\right)\right)\right)}} \cdot \sin th \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left({\sin kx}^{2}\right)\right)}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left({\sin kx}^{2}\right)\right)}} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left(\sin kx \cdot \sin kx\right)\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left(\sin kx \cdot \sin kx\right)\right)}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left(\sin kx \cdot \sin kx\right)\right)}} \cdot \sin th \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)\right)}} \cdot \sin th \]

      14. sub-negate N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(2 \cdot kx\right) - \frac{1}{2}\right)}} \cdot \sin th \]

      15. lower--.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(2 \cdot kx\right) - \frac{1}{2}\right)}} \cdot \sin th \]

      16. count-2-rev N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(kx + kx\right) - \frac{1}{2}\right)}} \cdot \sin th \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(kx + kx\right) - \frac{1}{2}\right)}} \cdot \sin th \]

      18. lower-cos.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(kx + kx\right) - \frac{1}{2}\right)}} \cdot \sin th \]

      19. lower-+.f64 26.9

          \[\leadsto \frac{ky}{\sqrt{-\left(0.5 \cdot \cos \left(kx + kx\right) - 0.5\right)}} \cdot \sin th \]

   6. Applied rewrites 26.9%

      \[\leadsto \frac{ky}{\sqrt{-\left(0.5 \cdot \cos \left(kx + kx\right) - 0.5\right)}} \cdot \sin th \]

   7. Taylor expanded in th around 0

      \[\leadsto \frac{ky}{\sqrt{-\left(\frac{1}{2} \cdot \cos \left(kx + kx\right) - \frac{1}{2}\right)}} \cdot \color{blue}{th} \]
   8. Step-by-step derivation

   9. Applied rewrites 14.8%

      \[\leadsto \frac{ky}{\sqrt{-\left(0.5 \cdot \cos \left(kx + kx\right) - 0.5\right)}} \cdot \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 16.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(ky \cdot \frac{1}{kx}\right) \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (* ky (/ 1.0 kx)) (sin th)))

C

double code(double kx, double ky, double th) {
	return (ky * (1.0 / 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 * (1.0d0 / kx)) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (ky * (1.0 / kx)) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (ky * (1.0 / kx)) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(ky * Float64(1.0 / kx)) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (ky * (1.0 / kx)) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(ky * N[(1.0 / kx), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in ky around 0

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. lower-sin.f64 36.7

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

4. Applied rewrites 36.7%

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
6. Step-by-step derivation

   1. lower-/.f64 16.8

      \[\leadsto \frac{ky}{kx} \cdot \sin th \]

7. Applied rewrites 16.8%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{ky}{kx} \cdot \sin th \]

   2. mult-flip N/A

      \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{kx}}\right) \cdot \sin th \]

   3. lower-*.f64 N/A

      \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{kx}}\right) \cdot \sin th \]

   4. lower-/.f64 16.8

      \[\leadsto \left(ky \cdot \frac{1}{kx}\right) \cdot \sin th \]

9. Applied rewrites 16.8%

   \[\leadsto \left(ky \cdot \frac{1}{\color{blue}{kx}}\right) \cdot \sin th \]
10. Add Preprocessing

Alternative 25: 16.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{ky}{kx} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ ky kx) (sin th)))

C

double code(double kx, double ky, double th) {
	return (ky / 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 / kx) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (ky / kx) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (ky / kx) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(ky / kx) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (ky / kx) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in ky around 0

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. lower-sin.f64 36.7

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

4. Applied rewrites 36.7%

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
6. Step-by-step derivation

   1. lower-/.f64 16.8

      \[\leadsto \frac{ky}{kx} \cdot \sin th \]

7. Applied rewrites 16.8%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
8. Add Preprocessing

Alternative 26: 13.9% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \frac{ky}{kx} \cdot th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ ky kx) th))

C

double code(double kx, double ky, double th) {
	return (ky / 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 / kx) * th
end function

Java

public static double code(double kx, double ky, double th) {
	return (ky / kx) * th;
}

Python

def code(kx, ky, th):
	return (ky / kx) * th

Julia

function code(kx, ky, th)
	return Float64(Float64(ky / kx) * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (ky / kx) * th;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(ky / kx), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in ky around 0

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   4. lower-sin.f64 36.7

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

4. Applied rewrites 36.7%

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
6. Step-by-step derivation

   1. lower-/.f64 16.8

      \[\leadsto \frac{ky}{kx} \cdot \sin th \]

7. Applied rewrites 16.8%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]

8. Taylor expanded in th around 0

   \[\leadsto \frac{ky}{kx} \cdot \color{blue}{th} \]
9. Step-by-step derivation

10. Applied rewrites 13.9%

    \[\leadsto \frac{ky}{kx} \cdot \color{blue}{th} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025144 
(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)))