Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 6.9s

Alternatives: 21

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: 79.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 5 regimes

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

   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. 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. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.0

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

   8. Applied rewrites 51.0%

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

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

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   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%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

Alternative 4: 78.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 5 regimes

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

   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. 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. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.0

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

   8. Applied rewrites 51.0%

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

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

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   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. 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. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 47.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

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

   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%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

Alternative 5: 78.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky)))
        (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)
       (* (sin ky) (/ th t_1))
       (if (<= t_3 0.45)
         (* (/ (sin ky) (fabs (sin kx))) (sin th))
         (if (<= t_3 0.9999999998354784)
           (/ (* (sin ky) th) t_1)
           (* (/ ky (hypot ky kx)) (sin th))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky));
	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 = sin(ky) * (th / t_1);
	elseif (t_3 <= 0.45)
		tmp = (sin(ky) / abs(sin(kx))) * sin(th);
	elseif (t_3 <= 0.9999999998354784)
		tmp = (sin(ky) * th) / t_1;
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 5 regimes

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

   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. 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. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.0

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

   8. Applied rewrites 51.0%

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

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

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   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. 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. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 47.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

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

   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%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

Alternative 6: 76.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.45], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999998354784], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -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-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 kx around 0

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

   6. Applied rewrites 58.3%

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

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

   1. Initial program 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. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.0

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

   8. Applied rewrites 51.0%

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

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

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   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. 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. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 47.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

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

   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%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

Alternative 7: 76.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.45], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999998354784], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -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-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 kx around 0

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

   6. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 54.7

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

      7. lift-hypot.f64 N/A

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

      8. sqrt-fabs-rev N/A

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

      9. lift-hypot.f64 N/A

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

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

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

      11. lift-hypot.f64 N/A

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

      12. lift-hypot.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. +-commutative N/A

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

   8. Applied rewrites 54.7%

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

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

   1. Initial program 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. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 51.0

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

   8. Applied rewrites 51.0%

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

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

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   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. 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. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 47.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

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

   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%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

Alternative 8: 76.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 0.45], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999998354784], t$95$2, N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -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-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 kx around 0

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

   6. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 54.7

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

      7. lift-hypot.f64 N/A

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

      8. sqrt-fabs-rev N/A

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

      9. lift-hypot.f64 N/A

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

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

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

      11. lift-hypot.f64 N/A

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

      12. lift-hypot.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. +-commutative N/A

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

   8. Applied rewrites 54.7%

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

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

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 47.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

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

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

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   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%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

Alternative 9: 68.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 47.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.5

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

   8. Applied rewrites 47.5%

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

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

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   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%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

Alternative 10: 64.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -1.0)
     (* (/ (sin ky) (hypot (sin ky) kx)) th)
     (if (<= t_1 0.708)
       (* (/ (sin ky) (fabs (sin kx))) (sin 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 tmp;
	if (t_1 <= -1.0) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * th;
	} else if (t_1 <= 0.708) {
		tmp = (sin(ky) / fabs(sin(kx))) * sin(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.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 = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * th;
	} else if (t_1 <= 0.708) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(kx))) * Math.sin(th);
	} 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 = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * th
	elif t_1 <= 0.708:
		tmp = (math.sin(ky) / math.fabs(math.sin(kx))) * math.sin(th)
	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(Float64(sin(ky) / hypot(sin(ky), kx)) * th);
	elseif (t_1 <= 0.708)
		tmp = Float64(Float64(sin(ky) / abs(sin(kx))) * sin(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) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = (sin(ky) / hypot(sin(ky), kx)) * th;
	elseif (t_1 <= 0.708)
		tmp = (sin(ky) / abs(sin(kx))) * sin(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[(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[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.708], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $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-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 kx around 0

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 33.6%

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

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

   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%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   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 11: 62.8% accurate, 0.4× speedup

Math

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

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 -1.0)
     (* (/ (sin ky) (hypot (sin ky) kx)) th)
     (if (<= t_1 0.708)
       (* (sin ky) (/ (sin th) (fabs (sin kx))))
       (* (/ 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 = (sin(ky) / hypot(sin(ky), kx)) * th;
	} else if (t_1 <= 0.708) {
		tmp = sin(ky) * (sin(th) / fabs(sin(kx)));
	} 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 = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * th;
	} else if (t_1 <= 0.708) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} 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 = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * th
	elif t_1 <= 0.708:
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(kx)))
	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(Float64(sin(ky) / hypot(sin(ky), kx)) * th);
	elseif (t_1 <= 0.708)
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(kx))));
	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 = (sin(ky) / hypot(sin(ky), kx)) * th;
	elseif (t_1 <= 0.708)
		tmp = sin(ky) * (sin(th) / abs(sin(kx)));
	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[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.708], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $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 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-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 kx around 0

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 33.6%

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

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

   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%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 40.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   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 12: 62.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}} \leq -0.12:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \frac{1}{\sqrt{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

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)
     (* (* th (sin ky)) (/ 1.0 (sqrt t_1)))
     (* (/ 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 = (th * sin(ky)) * (1.0 / sqrt(t_1));
	} 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 = (th * Math.sin(ky)) * (1.0 / Math.sqrt(t_1));
	} 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 = (th * math.sin(ky)) * (1.0 / math.sqrt(t_1))
	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(th * sin(ky)) * Float64(1.0 / sqrt(t_1)));
	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 = (th * sin(ky)) * (1.0 / sqrt(t_1));
	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[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[t$95$1], $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))))) < -0.12

   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. mult-flip N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 92.0

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

      11. lift-sqrt.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 95.9

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in kx around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 22.4

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

   9. Applied rewrites 22.4%

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

   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 13: 60.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) -0.01)
   (* (/ (sin ky) (hypot (sin ky) kx)) th)
   (* (/ 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) / hypot(sin(ky), kx)) * th;
	} 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 = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * th;
	} 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 = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * th
	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(Float64(sin(ky) / hypot(sin(ky), kx)) * th);
	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 = (sin(ky) / hypot(sin(ky), kx)) * th;
	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[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $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))))) < -0.0100000000000000002

   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-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 kx around 0

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 33.6%

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

   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 14: 59.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin kx\right|\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.12:\\ \;\;\;\;\sin ky \cdot \frac{th}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin th \cdot \frac{ky}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fabs (sin kx)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.12)
     (* (sin ky) (/ th t_1))
     (if (<= t_2 2e-6)
       (* (sin th) (/ ky t_1))
       (* (/ ky (hypot ky kx)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(sin(kx));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.12) {
		tmp = sin(ky) * (th / t_1);
	} else if (t_2 <= 2e-6) {
		tmp = sin(th) * (ky / t_1);
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.abs(Math.sin(kx));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.12) {
		tmp = Math.sin(ky) * (th / t_1);
	} else if (t_2 <= 2e-6) {
		tmp = Math.sin(th) * (ky / t_1);
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.fabs(math.sin(kx))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.12:
		tmp = math.sin(ky) * (th / t_1)
	elif t_2 <= 2e-6:
		tmp = math.sin(th) * (ky / t_1)
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = abs(sin(kx))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.12)
		tmp = Float64(sin(ky) * Float64(th / t_1));
	elseif (t_2 <= 2e-6)
		tmp = Float64(sin(th) * Float64(ky / t_1));
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(sin(kx));
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.12)
		tmp = sin(ky) * (th / t_1);
	elseif (t_2 <= 2e-6)
		tmp = sin(th) * (ky / t_1);
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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$2, -0.12], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-6], N[(N[Sin[th], $MachinePrecision] * N[(ky / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 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))))) < -0.12

   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%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 40.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 23.2%

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

   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))))) < 1.99999999999999991e-6

   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. rem-sqrt-square-rev N/A

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

      8. lower-fabs.f64 38.2

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

   6. Applied rewrites 38.2%

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

   if 1.99999999999999991e-6 < (/.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%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

Alternative 15: 51.2% 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: 49.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 3.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 3.05e+16)
   (* (/ ky (hypot ky kx)) (sin th))
   (* (sin ky) (/ th (fabs (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.05e+16) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = sin(ky) * (th / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 3.05e+16) {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(ky) * (th / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 3.05e+16:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	else:
		tmp = math.sin(ky) * (th / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 3.05e+16)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(sin(ky) * Float64(th / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 3.05e+16)
		tmp = (ky / hypot(ky, kx)) * sin(th);
	else
		tmp = sin(ky) * (th / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 3.05e+16], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 3.05e16

   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. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

   if 3.05e16 < kx

   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%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 40.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 43.4

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

   6. Applied rewrites 43.4%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 23.2%

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

Alternative 17: 46.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 100000000:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, -0.16666666666666666, th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 100000000.0)
   (*
    (/ ky (hypot ky (sin kx)))
    (fma (* (* th th) th) -0.16666666666666666 th))
   (* (/ ky (hypot ky kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 100000000.0) {
		tmp = (ky / hypot(ky, sin(kx))) * fma(((th * th) * th), -0.16666666666666666, th);
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 100000000.0)
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * fma(Float64(Float64(th * th) * th), -0.16666666666666666, th));
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 100000000.0], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666 + th), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1e8

   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. Taylor expanded in th around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 33.2

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

   12. Applied rewrites 33.2%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lift-pow.f64 N/A

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

       10. unpow2 N/A

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

       11. cube-unmult N/A

           \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \left({th}^{3} \cdot \frac{-1}{6} + 1 \cdot th\right) \]

       12. pow3 N/A

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

       13. unpow2 N/A

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

       14. lift-pow.f64 N/A

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

       15. *-lft-identity N/A

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

       16. lower-fma.f64 N/A

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

       17. lower-*.f64 33.2

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

       18. lift-pow.f64 N/A

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

       19. unpow2 N/A

           \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, \frac{-1}{6}, th\right) \]

       20. lower-*.f64 33.2

           \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \mathsf{fma}\left(\left(th \cdot th\right) \cdot th, -0.16666666666666666, th\right) \]

   14. Applied rewrites 33.2%

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

   if 1e8 < 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-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. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

Alternative 18: 42.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 350000000:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \left(th \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 350000000.0)
   (* (/ ky (hypot ky (sin kx))) (* th 1.0))
   (* (/ ky (hypot ky kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 350000000.0) {
		tmp = (ky / hypot(ky, sin(kx))) * (th * 1.0);
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 350000000.0) {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * (th * 1.0);
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 350000000.0:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * (th * 1.0)
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 350000000.0)
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * Float64(th * 1.0));
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 350000000.0)
		tmp = (ky / hypot(ky, sin(kx))) * (th * 1.0);
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 350000000.0], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 3.5e8

   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. Taylor expanded in th around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 33.2

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

   12. Applied rewrites 33.2%

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

   13. Taylor expanded in th around 0

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

   15. Applied rewrites 33.7%

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

   if 3.5e8 < 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-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. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.3%

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

Alternative 19: 42.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 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. Taylor expanded in kx around 0

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

12. Applied rewrites 46.3%

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

Alternative 20: 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 21: 13.2% accurate, 16.6× speedup

Math

\[\begin{array}{l} \\ \frac{ky}{kx} \cdot \left(th \cdot 1\right) \end{array} \]

FPCore

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

C

double code(double kx, double ky, double th) {
	return (ky / kx) * (th * 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (ky / kx) * (th * 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[(N[(ky / kx), $MachinePrecision] * N[(th * 1.0), $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. Taylor expanded in th around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 12.7

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

10. Applied rewrites 12.7%

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

11. Taylor expanded in th around 0

    \[\leadsto \frac{ky}{kx} \cdot \left(th \cdot 1\right) \]
12. Step-by-step derivation

13. Applied rewrites 13.2%

    \[\leadsto \frac{ky}{kx} \cdot \left(th \cdot 1\right) \]
14. 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)))