Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.6% → 99.7%

Time: 10.4s

Alternatives: 20

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. lift-*.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. lift-sin.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. associate-*l/ N/A

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

   12. associate-/l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

3. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.7

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

3. Applied rewrites 99.7%

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

Alternative 3: 78.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_1 \leq -0.9981:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.35:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\left|\sin kx\right|} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.9927:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))))
   (if (<= t_1 -0.9981)
     (* (/ (sin ky) (fabs (sin ky))) (sin th))
     (if (<= t_1 -0.35)
       t_2
       (if (<= t_1 5e-6)
         (* (/ (sin th) (fabs (sin kx))) (sin ky))
         (if (<= t_1 0.9927) t_2 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double tmp;
	if (t_1 <= -0.9981) {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	} else if (t_1 <= -0.35) {
		tmp = t_2;
	} else if (t_1 <= 5e-6) {
		tmp = (sin(th) / fabs(sin(kx))) * sin(ky);
	} else if (t_1 <= 0.9927) {
		tmp = t_2;
	} 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 t_2 = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (t_1 <= -0.9981) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(ky))) * Math.sin(th);
	} else if (t_1 <= -0.35) {
		tmp = t_2;
	} else if (t_1 <= 5e-6) {
		tmp = (Math.sin(th) / Math.abs(Math.sin(kx))) * Math.sin(ky);
	} else if (t_1 <= 0.9927) {
		tmp = t_2;
	} 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)))
	t_2 = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if t_1 <= -0.9981:
		tmp = (math.sin(ky) / math.fabs(math.sin(ky))) * math.sin(th)
	elif t_1 <= -0.35:
		tmp = t_2
	elif t_1 <= 5e-6:
		tmp = (math.sin(th) / math.fabs(math.sin(kx))) * math.sin(ky)
	elif t_1 <= 0.9927:
		tmp = t_2
	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))))
	t_2 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	tmp = 0.0
	if (t_1 <= -0.9981)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	elseif (t_1 <= -0.35)
		tmp = t_2;
	elseif (t_1 <= 5e-6)
		tmp = Float64(Float64(sin(th) / abs(sin(kx))) * sin(ky));
	elseif (t_1 <= 0.9927)
		tmp = t_2;
	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)));
	t_2 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (t_1 <= -0.9981)
		tmp = (sin(ky) / abs(sin(ky))) * sin(th);
	elseif (t_1 <= -0.35)
		tmp = t_2;
	elseif (t_1 <= 5e-6)
		tmp = (sin(th) / abs(sin(kx))) * sin(ky);
	elseif (t_1 <= 0.9927)
		tmp = t_2;
	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]}, 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, -0.9981], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.35], t$95$2, If[LessEqual[t$95$1, 5e-6], N[(N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9927], t$95$2, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99809999999999999

   1. Initial program 85.3%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. rem-sqrt-square N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift-sin.f64 90.8

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

   4. Applied rewrites 90.8%

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

   if -0.99809999999999999 < (/.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))))) < -0.34999999999999998 or 5.00000000000000041e-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))))) < 0.992700000000000027

   1. Initial program 99.3%

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

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

   4. Applied rewrites 50.5%

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

   if -0.34999999999999998 < (/.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))))) < 5.00000000000000041e-6

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lift-sin.f64 N/A

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

      6. lift-fabs.f64 94.4

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

   6. Applied rewrites 94.4%

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

   if 0.992700000000000027 < (/.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 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   3. Applied rewrites 99.4%

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

   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 5.0%

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

   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 13.0%

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

Alternative 4: 70.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.00049:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.00049)
   (*
    (/
     (*
      (fma
       (fma (* th th) 0.008333333333333333 -0.16666666666666666)
       (* th th)
       1.0)
      th)
     (hypot (sin kx) (sin ky)))
    (sin ky))
   (* (/ ky (hypot ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.00049) {
		tmp = ((fma(fma((th * th), 0.008333333333333333, -0.16666666666666666), (th * th), 1.0) * th) / hypot(sin(kx), sin(ky))) * sin(ky);
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.00049)
		tmp = Float64(Float64(Float64(fma(fma(Float64(th * th), 0.008333333333333333, -0.16666666666666666), Float64(th * th), 1.0) * th) / hypot(sin(kx), sin(ky))) * sin(ky));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.00049], N[(N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $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 th < 4.8999999999999998e-4

   1. Initial program 93.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)}}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{th}}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{th}}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      3. +-commutative N/A

         \[\leadsto \frac{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) \cdot {th}^{2} + 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}, {th}^{2}, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6} \cdot 1, {th}^{2}, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot 1, {th}^{2}, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({th}^{2} \cdot \frac{1}{120} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot 1, {th}^{2}, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({th}^{2} \cdot \frac{1}{120} + \frac{-1}{6} \cdot 1, {th}^{2}, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({th}^{2} \cdot \frac{1}{120} + \frac{-1}{6}, {th}^{2}, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left({th}^{2}, \frac{1}{120}, \frac{-1}{6}\right), {th}^{2}, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, \frac{1}{120}, \frac{-1}{6}\right), {th}^{2}, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, \frac{1}{120}, \frac{-1}{6}\right), {th}^{2}, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, \frac{1}{120}, \frac{-1}{6}\right), th \cdot th, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

      15. lower-*.f64 67.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

   6. Applied rewrites 67.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th}}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

   if 4.8999999999999998e-4 < th

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   3. Applied rewrites 99.6%

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

   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 50.8%

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

   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 64.1%

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

Alternative 5: 69.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.00049:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right)}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\right) \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.00049)
   (*
    (*
     (/ (fma (* th th) -0.16666666666666666 1.0) (hypot (sin kx) (sin ky)))
     th)
    (sin ky))
   (* (/ ky (hypot ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.00049) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) / hypot(sin(kx), sin(ky))) * th) * sin(ky);
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.00049)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) / hypot(sin(kx), sin(ky))) * th) * sin(ky));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.00049], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] * N[Sin[ky], $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 th < 4.8999999999999998e-4

   1. Initial program 93.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 67.4%

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

   if 4.8999999999999998e-4 < th

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   3. Applied rewrites 99.6%

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

   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 50.8%

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

   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 64.1%

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

Alternative 6: 66.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.00048:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.00048)
   (* (/ (sin ky) (hypot (sin kx) (sin ky))) th)
   (* (/ ky (hypot ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.00048) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.00048) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * th;
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.00048:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * 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 (th <= 0.00048)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.00048)
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.00048], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 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 th < 4.80000000000000012e-4

   1. Initial program 93.5%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 39.4

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 27.0%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 19.9%

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

   11. Taylor expanded in kx around inf

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

       1. lower-/.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. lower-hypot.f64 N/A

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

       6. lift-sin.f64 N/A

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

       7. lift-sin.f64 67.7

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

   13. Applied rewrites 67.7%

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

   if 4.80000000000000012e-4 < th

   1. Initial program 93.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   3. Applied rewrites 99.6%

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

   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 50.8%

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

   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 64.1%

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

Alternative 7: 66.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;ky \leq 4.8:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= ky 4.8)
     (* (/ (sin th) (hypot (sin kx) t_1)) t_1)
     (* (/ (sin ky) (fabs (sin ky))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (ky <= 4.8) {
		tmp = (sin(th) / hypot(sin(kx), t_1)) * t_1;
	} else {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (ky <= 4.8)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), t_1)) * t_1);
	else
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[ky, 4.8], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 4.79999999999999982

   1. Initial program 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 67.7

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)} \cdot \sin ky \]

   6. Applied rewrites 67.7%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.9

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \]

   9. Applied rewrites 68.9%

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

   if 4.79999999999999982 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. rem-sqrt-square N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift-sin.f64 59.5

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

   4. Applied rewrites 59.5%

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

Alternative 8: 66.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 130:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot ky\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 130.0)
   (* (/ (sin th) (hypot (sin kx) ky)) ky)
   (* (/ (sin ky) (fabs (sin ky))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 130.0) {
		tmp = (sin(th) / hypot(sin(kx), ky)) * ky;
	} else {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 130.0) {
		tmp = (Math.sin(th) / Math.hypot(Math.sin(kx), ky)) * ky;
	} else {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(ky))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 130.0:
		tmp = (math.sin(th) / math.hypot(math.sin(kx), ky)) * ky
	else:
		tmp = (math.sin(ky) / math.fabs(math.sin(ky))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 130.0)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), ky)) * ky);
	else
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 130.0)
		tmp = (sin(th) / hypot(sin(kx), ky)) * ky;
	else
		tmp = (sin(ky) / abs(sin(ky))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 130.0], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 130

   1. Initial program 91.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 67.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 74.3%

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

   if 130 < ky

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. rem-sqrt-square N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift-sin.f64 59.4

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

   4. Applied rewrites 59.4%

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

Alternative 9: 66.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= -0.02) {
		tmp = (sin(th) * sin(ky)) / fabs(sin(ky));
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= -0.02) {
		tmp = (Math.sin(th) * Math.sin(ky)) / Math.abs(Math.sin(ky));
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= -0.02:
		tmp = (math.sin(th) * math.sin(ky)) / math.fabs(math.sin(ky))
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.02)
		tmp = Float64(Float64(sin(th) * sin(ky)) / abs(sin(ky)));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.02)
		tmp = (sin(th) * sin(ky)) / abs(sin(ky));
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 90.4%

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

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. unpow2 N/A

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

      7. rem-sqrt-square N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift-sin.f64 63.0

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

   4. Applied rewrites 63.0%

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

   if -0.0200000000000000004 < (/.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 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   3. Applied rewrites 99.6%

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

   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 63.0%

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

   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 73.2%

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

Alternative 10: 60.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) -0.02)
   (* (/ (sin ky) (fabs (sin ky))) 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)))) <= -0.02) {
		tmp = (sin(ky) / fabs(sin(ky))) * th;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= -0.02) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(ky))) * 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)))) <= -0.02:
		tmp = (math.sin(ky) / math.fabs(math.sin(ky))) * 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)))) <= -0.02)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.02)
		tmp = (sin(ky) / abs(sin(ky))) * 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], -0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $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))))) < -0.0200000000000000004

   1. Initial program 90.4%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 8.5

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

   4. Applied rewrites 8.5%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 5.4%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 5.7%

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

   11. Taylor expanded in kx around 0

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

       1. lower-/.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. pow2 N/A

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

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

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

       5. lower-fabs.f64 N/A

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

       6. lift-sin.f64 33.1

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

   13. Applied rewrites 33.1%

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

   if -0.0200000000000000004 < (/.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 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   3. Applied rewrites 99.6%

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

   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 63.0%

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

   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 73.2%

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

Alternative 11: 54.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.02)
     (* (/ (sin ky) (fabs (sin ky))) th)
     (if (<= t_1 5e-6)
       (*
        (/ (* (fma (* ky ky) -0.16666666666666666 1.0) ky) (fabs (sin kx)))
        (sin th))
       (if (<= t_1 2.0)
         (* (sin th) (/ ky (sqrt (fma ky ky (* kx kx)))))
         (*
          (/
           ky
           (fabs
            (*
             (fma
              (fma 0.008333333333333333 (* kx kx) -0.16666666666666666)
              (* kx kx)
              1.0)
             kx)))
          (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = (sin(ky) / fabs(sin(ky))) * th;
	} else if (t_1 <= 5e-6) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / fabs(sin(kx))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th) * (ky / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = (ky / fabs((fma(fma(0.008333333333333333, (kx * kx), -0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * th);
	elseif (t_1 <= 5e-6)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / abs(sin(kx))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = Float64(sin(th) * Float64(ky / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(Float64(ky / abs(Float64(fma(fma(0.008333333333333333, Float64(kx * kx), -0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 5e-6], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Abs[N[(N[(N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 90.4%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 8.5

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

   4. Applied rewrites 8.5%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 5.4%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 5.7%

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

   11. Taylor expanded in kx around 0

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

       1. lower-/.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. pow2 N/A

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

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

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

       5. lower-fabs.f64 N/A

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

       6. lift-sin.f64 33.1

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

   13. Applied rewrites 33.1%

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

   if -0.0200000000000000004 < (/.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))))) < 5.00000000000000041e-6

   1. Initial program 99.4%

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

   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. unpow2 N/A

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

      2. rem-sqrt-square N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift-sin.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.8

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\left|\sin kx\right|} \cdot \sin th \]

   7. Applied rewrites 98.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\left|\sin kx\right|} \cdot \sin th \]

   if 5.00000000000000041e-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))))) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 29.6%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-sin.f64 29.6

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 29.6

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

   12. Applied rewrites 29.6%

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

   if 2 < (/.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 2.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\left|kx \cdot \left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right|} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{ky}{\left|\left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right) \cdot kx\right|} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky}{\left|\left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right) \cdot kx\right|} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{ky}{\left|\left({kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right) + 1\right) \cdot kx\right|} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{ky}{\left|\left(\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right) \cdot {kx}^{2} + 1\right) \cdot kx\right|} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      6. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6} \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \frac{-1}{6} \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \frac{-1}{6}, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {kx}^{2}, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      11. pow2 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      13. pow2 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]

      14. lift-*.f64 54.1

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]

   7. Applied rewrites 54.1%

      \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{ky}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.02)
     (* (/ (sin ky) (fabs (sin ky))) th)
     (if (<= t_1 5e-6)
       (* (/ ky (fabs (sin kx))) (sin th))
       (if (<= t_1 2.0)
         (* (sin th) (/ ky (sqrt (fma ky ky (* kx kx)))))
         (*
          (/
           ky
           (fabs
            (*
             (fma
              (fma 0.008333333333333333 (* kx kx) -0.16666666666666666)
              (* kx kx)
              1.0)
             kx)))
          (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = (sin(ky) / fabs(sin(ky))) * th;
	} else if (t_1 <= 5e-6) {
		tmp = (ky / fabs(sin(kx))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th) * (ky / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = (ky / fabs((fma(fma(0.008333333333333333, (kx * kx), -0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * th);
	elseif (t_1 <= 5e-6)
		tmp = Float64(Float64(ky / abs(sin(kx))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = Float64(sin(th) * Float64(ky / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(Float64(ky / abs(Float64(fma(fma(0.008333333333333333, Float64(kx * kx), -0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 5e-6], N[(N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Abs[N[(N[(N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 90.4%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 8.5

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

   4. Applied rewrites 8.5%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 5.4%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 5.7%

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

   11. Taylor expanded in kx around 0

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

       1. lower-/.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. pow2 N/A

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

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

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

       5. lower-fabs.f64 N/A

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

       6. lift-sin.f64 33.1

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

   13. Applied rewrites 33.1%

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

   if -0.0200000000000000004 < (/.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))))) < 5.00000000000000041e-6

   1. Initial program 99.4%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 5.00000000000000041e-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))))) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 29.6%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-sin.f64 29.6

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 29.6

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

   12. Applied rewrites 29.6%

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

   if 2 < (/.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 2.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\left|kx \cdot \left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right|} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{ky}{\left|\left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right) \cdot kx\right|} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky}{\left|\left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right) \cdot kx\right|} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{ky}{\left|\left({kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right) + 1\right) \cdot kx\right|} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{ky}{\left|\left(\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right) \cdot {kx}^{2} + 1\right) \cdot kx\right|} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      6. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6} \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \frac{-1}{6} \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \frac{-1}{6}, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {kx}^{2}, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      11. pow2 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      13. pow2 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]

      14. lift-*.f64 54.1

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]

   7. Applied rewrites 54.1%

      \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 46.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{ky}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th\\ \end{array} \end{array} \]

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 5e-6)
     (* (/ ky (fabs (sin kx))) (sin th))
     (if (<= t_1 2.0)
       (* (sin th) (/ ky (sqrt (fma ky ky (* kx kx)))))
       (*
        (/
         ky
         (fabs
          (*
           (fma
            (fma 0.008333333333333333 (* kx kx) -0.16666666666666666)
            (* kx kx)
            1.0)
           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 <= 5e-6) {
		tmp = (ky / fabs(sin(kx))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th) * (ky / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = (ky / fabs((fma(fma(0.008333333333333333, (kx * kx), -0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 5e-6)
		tmp = Float64(Float64(ky / abs(sin(kx))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = Float64(sin(th) * Float64(ky / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(Float64(ky / abs(Float64(fma(fma(0.008333333333333333, Float64(kx * kx), -0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-6], N[(N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Abs[N[(N[(N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision]], $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))))) < 5.00000000000000041e-6

   1. Initial program 95.0%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 54.5

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

   4. Applied rewrites 54.5%

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

   if 5.00000000000000041e-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))))) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 29.6%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-sin.f64 29.6

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 29.6

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

   12. Applied rewrites 29.6%

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

   if 2 < (/.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 2.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\left|kx \cdot \left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right|} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{ky}{\left|\left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right) \cdot kx\right|} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky}{\left|\left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right) \cdot kx\right|} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{ky}{\left|\left({kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right) + 1\right) \cdot kx\right|} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{ky}{\left|\left(\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right) \cdot {kx}^{2} + 1\right) \cdot kx\right|} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      6. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6} \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \frac{-1}{6} \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \frac{-1}{6}, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {kx}^{2}, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      11. pow2 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      13. pow2 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]

      14. lift-*.f64 54.1

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]

   7. Applied rewrites 54.1%

      \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 45.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th\\ \end{array} \end{array} \]

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 5e-6)
     (/ (* (sin th) ky) (fabs (sin kx)))
     (if (<= t_1 2.0)
       (* (sin th) (/ ky (sqrt (fma ky ky (* kx kx)))))
       (*
        (/
         ky
         (fabs
          (*
           (fma
            (fma 0.008333333333333333 (* kx kx) -0.16666666666666666)
            (* kx kx)
            1.0)
           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 <= 5e-6) {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = sin(th) * (ky / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = (ky / fabs((fma(fma(0.008333333333333333, (kx * kx), -0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 5e-6)
		tmp = Float64(Float64(sin(th) * ky) / abs(sin(kx)));
	elseif (t_1 <= 2.0)
		tmp = Float64(sin(th) * Float64(ky / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(Float64(ky / abs(Float64(fma(fma(0.008333333333333333, Float64(kx * kx), -0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-6], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Abs[N[(N[(N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision]], $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))))) < 5.00000000000000041e-6

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. pow2 N/A

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

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

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

      7. lift-sin.f64 N/A

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

      8. lift-fabs.f64 52.4

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

   6. Applied rewrites 52.4%

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

   if 5.00000000000000041e-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))))) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 29.6%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-sin.f64 29.6

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 29.6

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

   12. Applied rewrites 29.6%

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

   if 2 < (/.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 2.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\left|kx \cdot \left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)\right|} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{ky}{\left|\left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right) \cdot kx\right|} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky}{\left|\left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right) \cdot kx\right|} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{ky}{\left|\left({kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right) + 1\right) \cdot kx\right|} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{ky}{\left|\left(\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right) \cdot {kx}^{2} + 1\right) \cdot kx\right|} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      6. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6} \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \frac{-1}{6} \cdot 1, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} + \frac{-1}{6}, {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {kx}^{2}, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      11. pow2 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), {kx}^{2}, 1\right) \cdot kx\right|} \cdot \sin th \]

      13. pow2 N/A

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, kx \cdot kx, \frac{-1}{6}\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]

      14. lift-*.f64 54.1

          \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]

   7. Applied rewrites 54.1%

      \[\leadsto \frac{ky}{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right|} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 37.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.84:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.84)
   (* (sin th) (/ ky (sqrt (fma ky ky (* kx kx)))))
   (* (/ ky (sqrt (- 0.5 (* 0.5 (cos (+ kx kx)))))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.84) {
		tmp = sin(th) * (ky / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = (ky / sqrt((0.5 - (0.5 * cos((kx + kx)))))) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.84)
		tmp = Float64(sin(th) * Float64(ky / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(Float64(ky / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx)))))) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.84], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.839999999999999969

   1. Initial program 91.8%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.5

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 32.1%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 40.4%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-sin.f64 40.4

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 40.4

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

   12. Applied rewrites 40.4%

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

   if 0.839999999999999969 < kx

   1. Initial program 99.4%

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

   2. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. cos-2 N/A

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

      5. cos-sum N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 59.6

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

   4. Applied rewrites 59.6%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 31.4%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 26.9%

       \[\leadsto \frac{\color{blue}{ky}}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 37.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.84:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\left|\sin kx\right|} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.84)
   (* (sin th) (/ ky (sqrt (fma ky ky (* kx kx)))))
   (* (/ ky (fabs (sin kx))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.84) {
		tmp = sin(th) * (ky / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = (ky / fabs(sin(kx))) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.84)
		tmp = Float64(sin(th) * Float64(ky / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(Float64(ky / abs(sin(kx))) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.84], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.839999999999999969

   1. Initial program 91.8%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.5

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 32.1%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 40.4%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-sin.f64 40.4

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-fma.f64 40.4

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

   12. Applied rewrites 40.4%

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

   if 0.839999999999999969 < kx

   1. Initial program 99.4%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 51.2

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 27.0%

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

Alternative 17: 26.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{ky}{\left|kx\right|} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{ky}{\left|\sin kx\right|} \cdot th\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{ky}{\sqrt{kx \cdot kx + {ky}^{2}}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ ky (fabs kx)) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 2e-121)
     t_1
     (if (<= t_2 5e-6)
       (* (/ ky (fabs (sin kx))) th)
       (if (<= t_2 2.0)
         (*
          (/ ky (sqrt (+ (* kx kx) (pow ky 2.0))))
          (* (fma (* th th) -0.16666666666666666 1.0) th))
         t_1)))))

C

double code(double kx, double ky, double th) {
	double t_1 = (ky / fabs(kx)) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 2e-121) {
		tmp = t_1;
	} else if (t_2 <= 5e-6) {
		tmp = (ky / fabs(sin(kx))) * th;
	} else if (t_2 <= 2.0) {
		tmp = (ky / sqrt(((kx * kx) + pow(ky, 2.0)))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(ky / abs(kx)) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 2e-121)
		tmp = t_1;
	elseif (t_2 <= 5e-6)
		tmp = Float64(Float64(ky / abs(sin(kx))) * th);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(ky / sqrt(Float64(Float64(kx * kx) + (ky ^ 2.0)))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $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, 2e-121], t$95$1, If[LessEqual[t$95$2, 5e-6], N[(N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(ky / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[Power[ky, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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))))) < 2e-121 or 2 < (/.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 90.1%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 29.1%

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

   if 2e-121 < (/.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))))) < 5.00000000000000041e-6

   1. Initial program 98.9%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 48.4%

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

   if 5.00000000000000041e-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))))) < 2

   1. Initial program 99.4%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 29.6%

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

   11. Taylor expanded in th around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 16.5

          \[\leadsto \frac{ky}{\sqrt{kx \cdot kx + {ky}^{2}}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   13. Applied rewrites 16.5%

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

Alternative 18: 23.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 6.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{ky}{\left|\sin kx\right|} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\left|kx\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 6.8e+26)
   (* (/ ky (fabs (sin kx))) th)
   (* (/ ky (fabs kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 6.8e+26) {
		tmp = (ky / fabs(sin(kx))) * th;
	} else {
		tmp = (ky / fabs(kx)) * sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 6.8d+26) then
        tmp = (ky / abs(sin(kx))) * th
    else
        tmp = (ky / abs(kx)) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 6.8e+26) {
		tmp = (ky / Math.abs(Math.sin(kx))) * th;
	} else {
		tmp = (ky / Math.abs(kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 6.8e+26:
		tmp = (ky / math.fabs(math.sin(kx))) * th
	else:
		tmp = (ky / math.fabs(kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 6.8e+26)
		tmp = Float64(Float64(ky / abs(sin(kx))) * th);
	else
		tmp = Float64(Float64(ky / abs(kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 6.8e+26)
		tmp = (ky / abs(sin(kx))) * th;
	else
		tmp = (ky / abs(kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 6.8e+26], N[(N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 6.8000000000000005e26

   1. Initial program 93.5%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 39.3

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 26.3%

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

   if 6.8000000000000005e26 < th

   1. Initial program 93.9%

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

   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. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 38.1

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

   4. Applied rewrites 38.1%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 15.8%

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

Alternative 19: 20.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{ky}{\left|\sin kx\right|} \cdot th \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

2. Taylor expanded in ky around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. rem-sqrt-square N/A

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

   4. lower-fabs.f64 N/A

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

   5. lift-sin.f64 39.0

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

4. Applied rewrites 39.0%

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

5. Taylor expanded in th around 0

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

7. Applied rewrites 20.8%

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

Alternative 20: 15.7% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

2. Taylor expanded in ky around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. rem-sqrt-square N/A

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

   4. lower-fabs.f64 N/A

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

   5. lift-sin.f64 39.0

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

4. Applied rewrites 39.0%

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

5. Taylor expanded in th around 0

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

7. Applied rewrites 20.8%

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

8. Taylor expanded in kx around 0

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

10. Applied rewrites 15.7%

    \[\leadsto \frac{ky}{\left|kx\right|} \cdot th \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(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)))