Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 7.7s

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: 94.0% 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 94.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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 94.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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 93.9

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

   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: 76.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \sqrt{t\_1 + t\_2}\\ t_4 := \frac{\sin ky}{t\_3}\\ \mathbf{if}\;t\_4 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky\\ \mathbf{elif}\;t\_4 \leq 0.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9999999998354784:\\ \;\;\;\;\frac{th \cdot \sin ky}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (sqrt (+ t_1 t_2)))
        (t_4 (/ (sin ky) t_3)))
   (if (<= t_4 -0.98)
     (* (/ (sin ky) (sqrt t_2)) (sin th))
     (if (<= t_4 -0.2)
       (* (/ th (hypot (sin ky) (sin kx))) (sin ky))
       (if (<= t_4 0.45)
         (* (/ (sin ky) (sqrt t_1)) (sin th))
         (if (<= t_4 0.9999999998354784)
           (/ (* th (sin ky)) t_3)
           (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sqrt((t_1 + t_2));
	double t_4 = sin(ky) / t_3;
	double tmp;
	if (t_4 <= -0.98) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_4 <= -0.2) {
		tmp = (th / hypot(sin(ky), sin(kx))) * sin(ky);
	} else if (t_4 <= 0.45) {
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	} else if (t_4 <= 0.9999999998354784) {
		tmp = (th * sin(ky)) / t_3;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 2.0);
	double t_2 = Math.pow(Math.sin(ky), 2.0);
	double t_3 = Math.sqrt((t_1 + t_2));
	double t_4 = Math.sin(ky) / t_3;
	double tmp;
	if (t_4 <= -0.98) {
		tmp = (Math.sin(ky) / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_4 <= -0.2) {
		tmp = (th / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(ky);
	} else if (t_4 <= 0.45) {
		tmp = (Math.sin(ky) / Math.sqrt(t_1)) * Math.sin(th);
	} else if (t_4 <= 0.9999999998354784) {
		tmp = (th * Math.sin(ky)) / t_3;
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	t_2 = math.pow(math.sin(ky), 2.0)
	t_3 = math.sqrt((t_1 + t_2))
	t_4 = math.sin(ky) / t_3
	tmp = 0
	if t_4 <= -0.98:
		tmp = (math.sin(ky) / math.sqrt(t_2)) * math.sin(th)
	elif t_4 <= -0.2:
		tmp = (th / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(ky)
	elif t_4 <= 0.45:
		tmp = (math.sin(ky) / math.sqrt(t_1)) * math.sin(th)
	elif t_4 <= 0.9999999998354784:
		tmp = (th * math.sin(ky)) / t_3
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = sin(ky) ^ 2.0
	t_3 = sqrt(Float64(t_1 + t_2))
	t_4 = Float64(sin(ky) / t_3)
	tmp = 0.0
	if (t_4 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_4 <= -0.2)
		tmp = Float64(Float64(th / hypot(sin(ky), sin(kx))) * sin(ky));
	elseif (t_4 <= 0.45)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * sin(th));
	elseif (t_4 <= 0.9999999998354784)
		tmp = Float64(Float64(th * sin(ky)) / t_3);
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	t_2 = sin(ky) ^ 2.0;
	t_3 = sqrt((t_1 + t_2));
	t_4 = sin(ky) / t_3;
	tmp = 0.0;
	if (t_4 <= -0.98)
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	elseif (t_4 <= -0.2)
		tmp = (th / hypot(sin(ky), sin(kx))) * sin(ky);
	elseif (t_4 <= 0.45)
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	elseif (t_4 <= 0.9999999998354784)
		tmp = (th * sin(ky)) / t_3;
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.2], N[(N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.45], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9999999998354784], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 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))))) < -0.97999999999999998

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 51.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.450000000000000011

   1. Initial program 94.0%

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

   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 40.5

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

   4. Applied rewrites 40.5%

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

   if 0.450000000000000011 < (/.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))))) < 0.999999999835478381

   1. Initial program 94.0%

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

   2. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 46.4

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

   4. Applied rewrites 46.4%

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

   if 0.999999999835478381 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 4: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + t\_1}}\\ t_4 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\frac{th}{t\_4} \cdot \sin ky\\ \mathbf{elif}\;t\_3 \leq 0.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9999999998354784:\\ \;\;\;\;\frac{1}{\frac{t\_4}{\sin ky \cdot th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 t_1))))
        (t_4 (hypot (sin ky) (sin kx))))
   (if (<= t_3 -0.98)
     (* (/ (sin ky) (sqrt t_1)) (sin th))
     (if (<= t_3 -0.2)
       (* (/ th t_4) (sin ky))
       (if (<= t_3 0.45)
         (* (/ (sin ky) (sqrt t_2)) (sin th))
         (if (<= t_3 0.9999999998354784)
           (/ 1.0 (/ t_4 (* (sin ky) th)))
           (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + t_1));
	double t_4 = hypot(sin(ky), sin(kx));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	} else if (t_3 <= -0.2) {
		tmp = (th / t_4) * sin(ky);
	} else if (t_3 <= 0.45) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_3 <= 0.9999999998354784) {
		tmp = 1.0 / (t_4 / (sin(ky) * th));
	} else {
		tmp = (ky / hypot(ky, sin(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.pow(Math.sin(kx), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((t_2 + t_1));
	double t_4 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (Math.sin(ky) / Math.sqrt(t_1)) * Math.sin(th);
	} else if (t_3 <= -0.2) {
		tmp = (th / t_4) * Math.sin(ky);
	} else if (t_3 <= 0.45) {
		tmp = (Math.sin(ky) / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_3 <= 0.9999999998354784) {
		tmp = 1.0 / (t_4 / (Math.sin(ky) * th));
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.sin(ky) / math.sqrt((t_2 + t_1))
	t_4 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if t_3 <= -0.98:
		tmp = (math.sin(ky) / math.sqrt(t_1)) * math.sin(th)
	elif t_3 <= -0.2:
		tmp = (th / t_4) * math.sin(ky)
	elif t_3 <= 0.45:
		tmp = (math.sin(ky) / math.sqrt(t_2)) * math.sin(th)
	elif t_3 <= 0.9999999998354784:
		tmp = 1.0 / (t_4 / (math.sin(ky) * th))
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + t_1)))
	t_4 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * sin(th));
	elseif (t_3 <= -0.2)
		tmp = Float64(Float64(th / t_4) * sin(ky));
	elseif (t_3 <= 0.45)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_3 <= 0.9999999998354784)
		tmp = Float64(1.0 / Float64(t_4 / Float64(sin(ky) * th)));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(kx) ^ 2.0;
	t_3 = sin(ky) / sqrt((t_2 + t_1));
	t_4 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (t_3 <= -0.98)
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	elseif (t_3 <= -0.2)
		tmp = (th / t_4) * sin(ky);
	elseif (t_3 <= 0.45)
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	elseif (t_3 <= 0.9999999998354784)
		tmp = 1.0 / (t_4 / (sin(ky) * th));
	else
		tmp = (ky / hypot(ky, sin(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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[(th / t$95$4), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.45], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9999999998354784], N[(1.0 / N[(t$95$4 / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 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))))) < -0.97999999999999998

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 51.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.450000000000000011

   1. Initial program 94.0%

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

   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 40.5

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

   4. Applied rewrites 40.5%

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

   if 0.450000000000000011 < (/.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))))) < 0.999999999835478381

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 47.2%

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

   if 0.999999999835478381 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 5: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + t\_1}}\\ t_4 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_1}}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\frac{th}{t\_4} \cdot \sin ky\\ \mathbf{elif}\;t\_3 \leq 0.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9999999998354784:\\ \;\;\;\;\frac{1}{\frac{t\_4}{\sin ky \cdot th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 t_1))))
        (t_4 (hypot (sin ky) (sin kx))))
   (if (<= t_3 -0.98)
     (/ (* (sin ky) (sin th)) (sqrt t_1))
     (if (<= t_3 -0.2)
       (* (/ th t_4) (sin ky))
       (if (<= t_3 0.45)
         (* (/ (sin ky) (sqrt t_2)) (sin th))
         (if (<= t_3 0.9999999998354784)
           (/ 1.0 (/ t_4 (* (sin ky) th)))
           (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + t_1));
	double t_4 = hypot(sin(ky), sin(kx));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_1);
	} else if (t_3 <= -0.2) {
		tmp = (th / t_4) * sin(ky);
	} else if (t_3 <= 0.45) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_3 <= 0.9999999998354784) {
		tmp = 1.0 / (t_4 / (sin(ky) * th));
	} else {
		tmp = (ky / hypot(ky, sin(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.pow(Math.sin(kx), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((t_2 + t_1));
	double t_4 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (t_3 <= -0.98) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sqrt(t_1);
	} else if (t_3 <= -0.2) {
		tmp = (th / t_4) * Math.sin(ky);
	} else if (t_3 <= 0.45) {
		tmp = (Math.sin(ky) / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_3 <= 0.9999999998354784) {
		tmp = 1.0 / (t_4 / (Math.sin(ky) * th));
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.sin(ky) / math.sqrt((t_2 + t_1))
	t_4 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if t_3 <= -0.98:
		tmp = (math.sin(ky) * math.sin(th)) / math.sqrt(t_1)
	elif t_3 <= -0.2:
		tmp = (th / t_4) * math.sin(ky)
	elif t_3 <= 0.45:
		tmp = (math.sin(ky) / math.sqrt(t_2)) * math.sin(th)
	elif t_3 <= 0.9999999998354784:
		tmp = 1.0 / (t_4 / (math.sin(ky) * th))
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + t_1)))
	t_4 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (t_3 <= -0.98)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_1));
	elseif (t_3 <= -0.2)
		tmp = Float64(Float64(th / t_4) * sin(ky));
	elseif (t_3 <= 0.45)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_3 <= 0.9999999998354784)
		tmp = Float64(1.0 / Float64(t_4 / Float64(sin(ky) * th)));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(kx) ^ 2.0;
	t_3 = sin(ky) / sqrt((t_2 + t_1));
	t_4 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (t_3 <= -0.98)
		tmp = (sin(ky) * sin(th)) / sqrt(t_1);
	elseif (t_3 <= -0.2)
		tmp = (th / t_4) * sin(ky);
	elseif (t_3 <= 0.45)
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	elseif (t_3 <= 0.9999999998354784)
		tmp = 1.0 / (t_4 / (sin(ky) * th));
	else
		tmp = (ky / hypot(ky, sin(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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$3, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[(th / t$95$4), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.45], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9999999998354784], N[(1.0 / N[(t$95$4 / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 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))))) < -0.97999999999999998

   1. Initial program 94.0%

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

   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 42.3

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

   4. Applied rewrites 42.3%

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

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 51.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.450000000000000011

   1. Initial program 94.0%

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

   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 40.5

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

   4. Applied rewrites 40.5%

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

   if 0.450000000000000011 < (/.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))))) < 0.999999999835478381

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 47.2%

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

   if 0.999999999835478381 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 6: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.98:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\frac{th}{t\_1} \cdot \sin ky\\ \mathbf{elif}\;t\_3 \leq 0.45:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}}\\ \mathbf{elif}\;t\_3 \leq 0.9999999998354784:\\ \;\;\;\;\frac{1}{\frac{t\_1}{\sin ky \cdot th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.98)
     (/ (* (sin ky) (sin th)) (sqrt t_2))
     (if (<= t_3 -0.2)
       (* (/ th t_1) (sin ky))
       (if (<= t_3 0.45)
         (/ (sin th) (/ (fabs (sin kx)) (sin ky)))
         (if (<= t_3 0.9999999998354784)
           (/ 1.0 (/ t_1 (* (sin ky) th)))
           (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	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 <= -0.98) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_2);
	} else if (t_3 <= -0.2) {
		tmp = (th / t_1) * sin(ky);
	} else if (t_3 <= 0.45) {
		tmp = sin(th) / (fabs(sin(kx)) / sin(ky));
	} else if (t_3 <= 0.9999999998354784) {
		tmp = 1.0 / (t_1 / (sin(ky) * th));
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	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 <= -0.98) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sqrt(t_2);
	} else if (t_3 <= -0.2) {
		tmp = (th / t_1) * Math.sin(ky);
	} else if (t_3 <= 0.45) {
		tmp = Math.sin(th) / (Math.abs(Math.sin(kx)) / Math.sin(ky));
	} else if (t_3 <= 0.9999999998354784) {
		tmp = 1.0 / (t_1 / (Math.sin(ky) * th));
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	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 <= -0.98:
		tmp = (math.sin(ky) * math.sin(th)) / math.sqrt(t_2)
	elif t_3 <= -0.2:
		tmp = (th / t_1) * math.sin(ky)
	elif t_3 <= 0.45:
		tmp = math.sin(th) / (math.fabs(math.sin(kx)) / math.sin(ky))
	elif t_3 <= 0.9999999998354784:
		tmp = 1.0 / (t_1 / (math.sin(ky) * th))
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	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 <= -0.98)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_2));
	elseif (t_3 <= -0.2)
		tmp = Float64(Float64(th / t_1) * sin(ky));
	elseif (t_3 <= 0.45)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / sin(ky)));
	elseif (t_3 <= 0.9999999998354784)
		tmp = Float64(1.0 / Float64(t_1 / Float64(sin(ky) * th)));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	t_2 = sin(ky) ^ 2.0;
	t_3 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_2));
	tmp = 0.0;
	if (t_3 <= -0.98)
		tmp = (sin(ky) * sin(th)) / sqrt(t_2);
	elseif (t_3 <= -0.2)
		tmp = (th / t_1) * sin(ky);
	elseif (t_3 <= 0.45)
		tmp = sin(th) / (abs(sin(kx)) / sin(ky));
	elseif (t_3 <= 0.9999999998354784)
		tmp = 1.0 / (t_1 / (sin(ky) * th));
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $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, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[(th / t$95$1), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.45], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9999999998354784], N[(1.0 / N[(t$95$1 / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 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))))) < -0.97999999999999998

   1. Initial program 94.0%

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

   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 42.3

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

   4. Applied rewrites 42.3%

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

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 51.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.450000000000000011

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 94.0

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

      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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 85.1

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 76.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 76.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 31.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

   6. Applied rewrites 31.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 26.9%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. unpow-prod-down N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a-rev N/A

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

      9. pow1/2 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

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

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

      18. lower-fabs.f64 43.4

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

   10. Applied rewrites 43.4%

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

   if 0.450000000000000011 < (/.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))))) < 0.999999999835478381

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 47.2%

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

   if 0.999999999835478381 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 7: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{th}{t\_2} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.45:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 0.9999999998354784:\\ \;\;\;\;\frac{1}{\frac{t\_2}{\sin ky \cdot th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

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 (hypot (sin ky) (sin kx))))
   (if (<= t_1 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma (cos (+ ky ky)) -0.5 0.5))))
     (if (<= t_1 -0.2)
       (* (/ th t_2) (sin ky))
       (if (<= t_1 0.45)
         (/ (sin th) (/ (fabs (sin kx)) (sin ky)))
         (if (<= t_1 0.9999999998354784)
           (/ 1.0 (/ t_2 (* (sin ky) th)))
           (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = hypot(sin(ky), sin(kx));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(cos((ky + ky)), -0.5, 0.5)));
	} else if (t_1 <= -0.2) {
		tmp = (th / t_2) * sin(ky);
	} else if (t_1 <= 0.45) {
		tmp = sin(th) / (fabs(sin(kx)) / sin(ky));
	} else if (t_1 <= 0.9999999998354784) {
		tmp = 1.0 / (t_2 / (sin(ky) * th));
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * 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 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(cos(Float64(ky + ky)), -0.5, 0.5))));
	elseif (t_1 <= -0.2)
		tmp = Float64(Float64(th / t_2) * sin(ky));
	elseif (t_1 <= 0.45)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / sin(ky)));
	elseif (t_1 <= 0.9999999998354784)
		tmp = Float64(1.0 / Float64(t_2 / Float64(sin(ky) * th)));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return 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[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], N[(N[(th / t$95$2), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.45], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999998354784], N[(1.0 / N[(t$95$2 / N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 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))))) < -0.97999999999999998

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.6

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sqr-sin-a-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

   6. Applied rewrites 32.0%

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

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 51.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.450000000000000011

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 94.0

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

      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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 85.1

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 76.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 76.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 31.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

   6. Applied rewrites 31.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 26.9%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. unpow-prod-down N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a-rev N/A

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

      9. pow1/2 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

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

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

      18. lower-fabs.f64 43.4

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

   10. Applied rewrites 43.4%

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

   if 0.450000000000000011 < (/.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))))) < 0.999999999835478381

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 47.2%

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

   if 0.999999999835478381 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 8: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.45:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 0.9999999998354784:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

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 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma (cos (+ ky ky)) -0.5 0.5))))
     (if (<= t_1 -0.2)
       (* (/ th (hypot (sin ky) (sin kx))) (sin ky))
       (if (<= t_1 0.45)
         (/ (sin th) (/ (fabs (sin kx)) (sin ky)))
         (if (<= t_1 0.9999999998354784)
           (/ (* th (sin ky)) (hypot (sin kx) (sin ky)))
           (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(cos((ky + ky)), -0.5, 0.5)));
	} else if (t_1 <= -0.2) {
		tmp = (th / hypot(sin(ky), sin(kx))) * sin(ky);
	} else if (t_1 <= 0.45) {
		tmp = sin(th) / (fabs(sin(kx)) / sin(ky));
	} else if (t_1 <= 0.9999999998354784) {
		tmp = (th * sin(ky)) / hypot(sin(kx), sin(ky));
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * 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 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(cos(Float64(ky + ky)), -0.5, 0.5))));
	elseif (t_1 <= -0.2)
		tmp = Float64(Float64(th / hypot(sin(ky), sin(kx))) * sin(ky));
	elseif (t_1 <= 0.45)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / sin(ky)));
	elseif (t_1 <= 0.9999999998354784)
		tmp = Float64(Float64(th * sin(ky)) / hypot(sin(kx), sin(ky)));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return 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, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], N[(N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.45], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999998354784], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 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))))) < -0.97999999999999998

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.6

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sqr-sin-a-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

   6. Applied rewrites 32.0%

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

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 51.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.450000000000000011

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 94.0

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

      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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 85.1

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 76.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 76.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 31.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

   6. Applied rewrites 31.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 26.9%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. unpow-prod-down N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a-rev N/A

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

      9. pow1/2 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

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

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

      18. lower-fabs.f64 43.4

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

   10. Applied rewrites 43.4%

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

   if 0.450000000000000011 < (/.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))))) < 0.999999999835478381

   1. Initial program 94.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.1

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 96.1

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

   3. Applied rewrites 96.1%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.5%

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

   if 0.999999999835478381 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 9: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{th}{t\_2} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.45:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 0.9999999998354784:\\ \;\;\;\;\frac{\sin ky}{t\_2} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

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 (hypot (sin ky) (sin kx))))
   (if (<= t_1 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma (cos (+ ky ky)) -0.5 0.5))))
     (if (<= t_1 -0.2)
       (* (/ th t_2) (sin ky))
       (if (<= t_1 0.45)
         (/ (sin th) (/ (fabs (sin kx)) (sin ky)))
         (if (<= t_1 0.9999999998354784)
           (* (/ (sin ky) t_2) th)
           (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = hypot(sin(ky), sin(kx));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(cos((ky + ky)), -0.5, 0.5)));
	} else if (t_1 <= -0.2) {
		tmp = (th / t_2) * sin(ky);
	} else if (t_1 <= 0.45) {
		tmp = sin(th) / (fabs(sin(kx)) / sin(ky));
	} else if (t_1 <= 0.9999999998354784) {
		tmp = (sin(ky) / t_2) * th;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * 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 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(cos(Float64(ky + ky)), -0.5, 0.5))));
	elseif (t_1 <= -0.2)
		tmp = Float64(Float64(th / t_2) * sin(ky));
	elseif (t_1 <= 0.45)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / sin(ky)));
	elseif (t_1 <= 0.9999999998354784)
		tmp = Float64(Float64(sin(ky) / t_2) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return 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[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], N[(N[(th / t$95$2), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.45], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999998354784], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$2), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 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))))) < -0.97999999999999998

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.6

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sqr-sin-a-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

   6. Applied rewrites 32.0%

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

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 51.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.450000000000000011

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 94.0

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

      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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 85.1

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 76.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 76.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 31.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

   6. Applied rewrites 31.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 26.9%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. unpow-prod-down N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a-rev N/A

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

      9. pow1/2 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

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

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

      18. lower-fabs.f64 43.4

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

   10. Applied rewrites 43.4%

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

   if 0.450000000000000011 < (/.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))))) < 0.999999999835478381

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.0%

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

   if 0.999999999835478381 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 10: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.45:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 0.9999999998354784:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

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 ky) (sin kx))) (sin ky))))
   (if (<= t_1 -0.98)
     (* (sin ky) (/ (sin th) (sqrt (fma (cos (+ ky ky)) -0.5 0.5))))
     (if (<= t_1 -0.2)
       t_2
       (if (<= t_1 0.45)
         (/ (sin th) (/ (fabs (sin kx)) (sin ky)))
         (if (<= t_1 0.9999999998354784)
           t_2
           (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (th / hypot(sin(ky), sin(kx))) * sin(ky);
	double tmp;
	if (t_1 <= -0.98) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(cos((ky + ky)), -0.5, 0.5)));
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 0.45) {
		tmp = sin(th) / (fabs(sin(kx)) / sin(ky));
	} else if (t_1 <= 0.9999999998354784) {
		tmp = t_2;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * 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(ky), sin(kx))) * sin(ky))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(cos(Float64(ky + ky)), -0.5, 0.5))));
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 0.45)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / sin(ky)));
	elseif (t_1 <= 0.9999999998354784)
		tmp = t_2;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return 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[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 0.45], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999998354784], t$95$2, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 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))))) < -0.97999999999999998

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.6

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sqr-sin-a-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

   6. Applied rewrites 32.0%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.450000000000000011 < (/.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))))) < 0.999999999835478381

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 51.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.450000000000000011

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 94.0

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

      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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 85.1

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 76.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 76.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 31.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

   6. Applied rewrites 31.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 26.9%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. unpow-prod-down N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a-rev N/A

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

      9. pow1/2 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

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

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

      18. lower-fabs.f64 43.4

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

   10. Applied rewrites 43.4%

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

   if 0.999999999835478381 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 11: 64.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.01:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(\cos \left(ky + ky\right), -0.5, 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) -0.01)
   (* (sin ky) (/ (sin th) (sqrt (fma (cos (+ ky ky)) -0.5 0.5))))
   (* (/ ky (hypot ky (sin kx))) (sin th))))

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)))) <= -0.01) {
		tmp = sin(ky) * (sin(th) / sqrt(fma(cos((ky + ky)), -0.5, 0.5)));
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	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)))) <= -0.01)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(cos(Float64(ky + ky)), -0.5, 0.5))));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return 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], -0.01], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.6

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. sqr-sin-a-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

   6. Applied rewrites 32.0%

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

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 12: 64.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 0.708:\\ \;\;\;\;\frac{\sin th}{\frac{\left|\sin kx\right|}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

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 -1.0)
     (/ th (/ (hypot kx (sin ky)) (sin ky)))
     (if (<= t_1 0.708)
       (/ (sin th) (/ (fabs (sin kx)) (sin ky)))
       (* (/ ky (hypot ky (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = th / (hypot(kx, sin(ky)) / sin(ky));
	} else if (t_1 <= 0.708) {
		tmp = sin(th) / (fabs(sin(kx)) / sin(ky));
	} else {
		tmp = (ky / hypot(ky, sin(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 <= -1.0) {
		tmp = th / (Math.hypot(kx, Math.sin(ky)) / Math.sin(ky));
	} else if (t_1 <= 0.708) {
		tmp = Math.sin(th) / (Math.abs(Math.sin(kx)) / Math.sin(ky));
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(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 <= -1.0:
		tmp = th / (math.hypot(kx, math.sin(ky)) / math.sin(ky))
	elif t_1 <= 0.708:
		tmp = math.sin(th) / (math.fabs(math.sin(kx)) / math.sin(ky))
	else:
		tmp = (ky / math.hypot(ky, math.sin(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 <= -1.0)
		tmp = Float64(th / Float64(hypot(kx, sin(ky)) / sin(ky)));
	elseif (t_1 <= 0.708)
		tmp = Float64(sin(th) / Float64(abs(sin(kx)) / sin(ky)));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(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 <= -1.0)
		tmp = th / (hypot(kx, sin(ky)) / sin(ky));
	elseif (t_1 <= 0.708)
		tmp = sin(th) / (abs(sin(kx)) / sin(ky));
	else
		tmp = (ky / hypot(ky, sin(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, -1.0], N[(th / N[(N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.708], N[(N[Sin[th], $MachinePrecision] / N[(N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in kx around 0

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

   9. Applied rewrites 33.6%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.70799999999999996

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 94.0

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

      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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 85.1

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 76.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 76.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 31.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

   6. Applied rewrites 31.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 26.9%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. unpow-prod-down N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a-rev N/A

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

      9. pow1/2 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

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

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

      18. lower-fabs.f64 43.4

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

   10. Applied rewrites 43.4%

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

   if 0.70799999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 13: 62.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}} \leq -0.12:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0)))
   (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1))) -0.12)
     (* (/ (sin ky) (sqrt t_1)) th)
     (* (/ ky (hypot ky (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1))) <= -0.12) {
		tmp = (sin(ky) / sqrt(t_1)) * th;
	} else {
		tmp = (ky / hypot(ky, sin(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 tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1))) <= -0.12) {
		tmp = (Math.sin(ky) / Math.sqrt(t_1)) * th;
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))) <= -0.12:
		tmp = (math.sin(ky) / math.sqrt(t_1)) * th
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1))) <= -0.12)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1))) <= -0.12)
		tmp = (sin(ky) / sqrt(t_1)) * th;
	else
		tmp = (ky / hypot(ky, sin(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]}, If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.12], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.12

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 21.8%

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

   if -0.12 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 14: 60.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) -0.01)
   (/ th (/ (hypot kx (sin ky)) (sin ky)))
   (* (/ ky (hypot ky (sin kx))) (sin th))))

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)))) <= -0.01) {
		tmp = th / (hypot(kx, sin(ky)) / sin(ky));
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	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)))) <= -0.01) {
		tmp = th / (Math.hypot(kx, Math.sin(ky)) / Math.sin(ky));
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	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)))) <= -0.01:
		tmp = th / (math.hypot(kx, math.sin(ky)) / math.sin(ky))
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	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)))) <= -0.01)
		tmp = Float64(th / Float64(hypot(kx, sin(ky)) / sin(ky)));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	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)))) <= -0.01)
		tmp = th / (hypot(kx, sin(ky)) / sin(ky));
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	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], -0.01], N[(th / N[(N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in kx around 0

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

   9. Applied rewrites 33.6%

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

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.1%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.5%

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

Alternative 15: 59.6% 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 94.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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 50.1%

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

7. Taylor expanded in ky around 0

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

9. Applied rewrites 64.5%

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

Alternative 16: 42.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.225:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\sin kx\right|} \cdot \left(\sin th \cdot ky\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.225)
   (/ th (/ (hypot (sin kx) ky) ky))
   (* (/ 1.0 (fabs (sin kx))) (* (sin th) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.225) {
		tmp = th / (hypot(sin(kx), ky) / ky);
	} else {
		tmp = (1.0 / fabs(sin(kx))) * (sin(th) * ky);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.225) {
		tmp = th / (Math.hypot(Math.sin(kx), ky) / ky);
	} else {
		tmp = (1.0 / Math.abs(Math.sin(kx))) * (Math.sin(th) * ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.225:
		tmp = th / (math.hypot(math.sin(kx), ky) / ky)
	else:
		tmp = (1.0 / math.fabs(math.sin(kx))) * (math.sin(th) * ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.225)
		tmp = Float64(th / Float64(hypot(sin(kx), ky) / ky));
	else
		tmp = Float64(Float64(1.0 / abs(sin(kx))) * Float64(sin(th) * ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.225)
		tmp = th / (hypot(sin(kx), ky) / ky);
	else
		tmp = (1.0 / abs(sin(kx))) * (sin(th) * ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.225], N[(th / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.225000000000000006

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 27.6%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 33.7%

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

   if 0.225000000000000006 < th

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 94.0

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

      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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 85.1

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 76.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 76.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 26.6

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   8. Applied rewrites 23.6%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. unpow-prod-down N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a-rev N/A

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

      9. pow1/2 N/A

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

      10. sqr-sin-a-rev N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. pow2 N/A

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

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

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

      18. lower-fabs.f64 36.4

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

   10. Applied rewrites 36.4%

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

Alternative 17: 37.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 12.5:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 12.5)
   (/ th (/ (hypot (sin kx) ky) ky))
   (* (sin th) (/ ky (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 12.5) {
		tmp = th / (hypot(sin(kx), ky) / ky);
	} else {
		tmp = sin(th) * (ky / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 12.5) {
		tmp = th / (Math.hypot(Math.sin(kx), ky) / ky);
	} else {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 12.5:
		tmp = th / (math.hypot(math.sin(kx), ky) / ky)
	else:
		tmp = math.sin(th) * (ky / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 12.5)
		tmp = Float64(th / Float64(hypot(sin(kx), ky) / ky));
	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 (th <= 12.5)
		tmp = th / (hypot(sin(kx), ky) / ky);
	else
		tmp = sin(th) * (ky / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 12.5], N[(th / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 12.5

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 27.6%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 33.7%

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

   if 12.5 < th

   1. Initial program 94.0%

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

   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 35.3

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

   4. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   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 35.3

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sqr-sin-a-rev N/A

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

      10. lift-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. pow1/2 N/A

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

   6. Applied rewrites 24.3%

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

Alternative 18: 37.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 12.5:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 12.5)
   (/ th (/ (hypot (sin kx) ky) ky))
   (* (/ (sin th) (sin kx)) ky)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 12.5) {
		tmp = th / (hypot(sin(kx), ky) / ky);
	} else {
		tmp = (sin(th) / sin(kx)) * ky;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 12.5) {
		tmp = th / (Math.hypot(Math.sin(kx), ky) / ky);
	} else {
		tmp = (Math.sin(th) / Math.sin(kx)) * ky;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 12.5:
		tmp = th / (math.hypot(math.sin(kx), ky) / ky)
	else:
		tmp = (math.sin(th) / math.sin(kx)) * ky
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 12.5)
		tmp = Float64(th / Float64(hypot(sin(kx), ky) / ky));
	else
		tmp = Float64(Float64(sin(th) / sin(kx)) * ky);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 12.5)
		tmp = th / (hypot(sin(kx), ky) / ky);
	else
		tmp = (sin(th) / sin(kx)) * ky;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 12.5], N[(th / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 12.5

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 27.6%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 33.7%

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

   if 12.5 < th

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 94.0

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

      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-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sqr-sin-a N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-*.f64 85.1

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. lower--.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 76.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 76.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 26.6

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 24.4%

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

Alternative 19: 37.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{{ky}^{2}}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 2.4e-9)
   (/ th (/ (hypot (sin kx) ky) ky))
   (* (/ ky (sqrt (pow ky 2.0))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 2.4e-9) {
		tmp = th / (hypot(sin(kx), ky) / ky);
	} else {
		tmp = (ky / sqrt(pow(ky, 2.0))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 2.4e-9) {
		tmp = th / (Math.hypot(Math.sin(kx), ky) / ky);
	} else {
		tmp = (ky / Math.sqrt(Math.pow(ky, 2.0))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 2.4e-9:
		tmp = th / (math.hypot(math.sin(kx), ky) / ky)
	else:
		tmp = (ky / math.sqrt(math.pow(ky, 2.0))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 2.4e-9)
		tmp = Float64(th / Float64(hypot(sin(kx), ky) / ky));
	else
		tmp = Float64(Float64(ky / sqrt((ky ^ 2.0))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 2.4e-9)
		tmp = th / (hypot(sin(kx), ky) / ky);
	else
		tmp = (ky / sqrt((ky ^ 2.0))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 2.4e-9], N[(th / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Power[ky, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 2.4e-9

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 27.6%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 33.7%

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

   if 2.4e-9 < th

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 19.9%

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

Alternative 20: 35.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 4000000000000:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, ky\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 4000000000000.0)
   (/ th (/ (hypot (sin kx) ky) ky))
   (* (/ ky kx) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 4000000000000.0) {
		tmp = th / (hypot(sin(kx), ky) / ky);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 4000000000000.0) {
		tmp = th / (Math.hypot(Math.sin(kx), ky) / ky);
	} else {
		tmp = (ky / kx) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 4000000000000.0:
		tmp = th / (math.hypot(math.sin(kx), ky) / ky)
	else:
		tmp = (ky / kx) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 4000000000000.0)
		tmp = Float64(th / Float64(hypot(sin(kx), ky) / ky));
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 4000000000000.0)
		tmp = th / (hypot(sin(kx), ky) / ky);
	else
		tmp = (ky / kx) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 4000000000000.0], N[(th / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 4e12

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 94.0

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

      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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 27.6%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 33.7%

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

   if 4e12 < th

   1. Initial program 94.0%

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

   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 35.3

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

   4. Applied rewrites 35.3%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.2

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

   7. Applied rewrites 16.2%

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

Alternative 21: 18.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.7 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{\frac{kx}{ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot th}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.7e+26)
   (* (/ 1.0 (/ kx ky)) (sin th))
   (/ (* ky th) (sqrt (- 0.5 (* 0.5 (cos (* 2.0 kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.7e+26) {
		tmp = (1.0 / (kx / ky)) * sin(th);
	} else {
		tmp = (ky * th) / sqrt((0.5 - (0.5 * cos((2.0 * kx)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 1.7d+26) then
        tmp = (1.0d0 / (kx / ky)) * sin(th)
    else
        tmp = (ky * th) / sqrt((0.5d0 - (0.5d0 * cos((2.0d0 * kx)))))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.7e+26) {
		tmp = (1.0 / (kx / ky)) * Math.sin(th);
	} else {
		tmp = (ky * th) / Math.sqrt((0.5 - (0.5 * Math.cos((2.0 * kx)))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.7e+26:
		tmp = (1.0 / (kx / ky)) * math.sin(th)
	else:
		tmp = (ky * th) / math.sqrt((0.5 - (0.5 * math.cos((2.0 * kx)))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.7e+26)
		tmp = Float64(Float64(1.0 / Float64(kx / ky)) * sin(th));
	else
		tmp = Float64(Float64(ky * th) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.7e+26)
		tmp = (1.0 / (kx / ky)) * sin(th);
	else
		tmp = (ky * th) / sqrt((0.5 - (0.5 * cos((2.0 * kx)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.7e+26], N[(N[(1.0 / N[(kx / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.7000000000000001e26

   1. Initial program 94.0%

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

   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 35.3

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

   4. Applied rewrites 35.3%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.2

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

   7. Applied rewrites 16.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 16.2

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

   9. Applied rewrites 16.2%

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

   if 1.7000000000000001e26 < kx

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      3. lower-+.f64 94.0

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      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-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)} + {\sin kx}^{2}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      12. lower-*.f64 85.1

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right) + {\sin kx}^{2}}} \cdot \sin th \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      16. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      17. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      18. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      20. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

      21. lower-*.f64 76.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   3. Applied rewrites 76.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      8. lower-*.f64 26.6

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \]

   6. Applied rewrites 26.6%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]

   7. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 14.4%

      \[\leadsto \frac{ky \cdot th}{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot kx\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 16.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{kx}{ky}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ 1.0 (/ kx ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	return (1.0 / (kx / ky)) * 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 = (1.0d0 / (kx / ky)) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (1.0 / (kx / ky)) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (1.0 / (kx / ky)) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(1.0 / Float64(kx / ky)) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (1.0 / (kx / ky)) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(1.0 / N[(kx / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

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 35.3

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

4. Applied rewrites 35.3%

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
6. Step-by-step derivation

   1. lower-/.f64 16.2

      \[\leadsto \frac{ky}{kx} \cdot \sin th \]

7. Applied rewrites 16.2%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{ky}{kx} \cdot \sin th \]

   2. div-flip N/A

      \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{2}{2}}{\frac{kx}{ky}} \cdot \sin th \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{2}{2}}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

   5. metadata-eval N/A

      \[\leadsto \frac{1}{\frac{kx}{ky}} \cdot \sin th \]

   6. lower-/.f64 16.2

      \[\leadsto \frac{1}{\frac{kx}{ky}} \cdot \sin th \]

9. Applied rewrites 16.2%

   \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]
10. Add Preprocessing

Alternative 23: 16.2% 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 94.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

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 35.3

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

4. Applied rewrites 35.3%

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
6. Step-by-step derivation

   1. lower-/.f64 16.2

      \[\leadsto \frac{ky}{kx} \cdot \sin th \]

7. Applied rewrites 16.2%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
8. Add Preprocessing

Alternative 24: 13.2% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{kx}{ky}} \cdot th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ 1.0 (/ kx ky)) th))

C

double code(double kx, double ky, double th) {
	return (1.0 / (kx / ky)) * th;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (1.0d0 / (kx / ky)) * th
end function

Java

public static double code(double kx, double ky, double th) {
	return (1.0 / (kx / ky)) * th;
}

Python

def code(kx, ky, th):
	return (1.0 / (kx / ky)) * th

Julia

function code(kx, ky, th)
	return Float64(Float64(1.0 / Float64(kx / ky)) * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (1.0 / (kx / ky)) * th;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(1.0 / N[(kx / ky), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

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 35.3

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

4. Applied rewrites 35.3%

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
6. Step-by-step derivation

   1. lower-/.f64 16.2

      \[\leadsto \frac{ky}{kx} \cdot \sin th \]

7. Applied rewrites 16.2%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]

8. Taylor expanded in th around 0

   \[\leadsto \frac{ky}{kx} \cdot \color{blue}{th} \]
9. Step-by-step derivation

10. Applied rewrites 13.2%

    \[\leadsto \frac{ky}{kx} \cdot \color{blue}{th} \]
11. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{ky}{kx} \cdot th \]

    2. div-flip N/A

       \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot th \]

    3. lower-/.f64 N/A

       \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot th \]

    4. lower-/.f64 13.2

       \[\leadsto \frac{1}{\frac{kx}{ky}} \cdot th \]

12. Applied rewrites 13.2%

    \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot th \]
13. Add Preprocessing

Alternative 25: 13.2% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{kx} \cdot ky\right) \cdot th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (* (/ 1.0 kx) ky) th))

C

double code(double kx, double ky, double th) {
	return ((1.0 / kx) * ky) * th;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = ((1.0d0 / kx) * ky) * th
end function

Java

public static double code(double kx, double ky, double th) {
	return ((1.0 / kx) * ky) * th;
}

Python

def code(kx, ky, th):
	return ((1.0 / kx) * ky) * th

Julia

function code(kx, ky, th)
	return Float64(Float64(Float64(1.0 / kx) * ky) * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = ((1.0 / kx) * ky) * th;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[(1.0 / kx), $MachinePrecision] * ky), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 94.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

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 35.3

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

4. Applied rewrites 35.3%

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
6. Step-by-step derivation

   1. lower-/.f64 16.2

      \[\leadsto \frac{ky}{kx} \cdot \sin th \]

7. Applied rewrites 16.2%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]

8. Taylor expanded in th around 0

   \[\leadsto \frac{ky}{kx} \cdot \color{blue}{th} \]
9. Step-by-step derivation

10. Applied rewrites 13.2%

    \[\leadsto \frac{ky}{kx} \cdot \color{blue}{th} \]
11. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{ky}{kx} \cdot th \]

    2. div-flip N/A

       \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot th \]

    3. associate-/r/ N/A

       \[\leadsto \left(\frac{1}{kx} \cdot ky\right) \cdot th \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{kx} \cdot ky\right) \cdot th \]

    5. lower-/.f64 13.2

       \[\leadsto \left(\frac{1}{kx} \cdot ky\right) \cdot th \]

12. Applied rewrites 13.2%

    \[\leadsto \left(\frac{1}{kx} \cdot ky\right) \cdot th \]
13. Add Preprocessing

Alternative 26: 13.2% 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 94.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

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 35.3

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

4. Applied rewrites 35.3%

   \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
6. Step-by-step derivation

   1. lower-/.f64 16.2

      \[\leadsto \frac{ky}{kx} \cdot \sin th \]

7. Applied rewrites 16.2%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]

8. Taylor expanded in th around 0

   \[\leadsto \frac{ky}{kx} \cdot \color{blue}{th} \]
9. Step-by-step derivation

10. Applied rewrites 13.2%

    \[\leadsto \frac{ky}{kx} \cdot \color{blue}{th} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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)))