Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.6%

Time: 8.2s

Alternatives: 24

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: 93.7% 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: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.7%

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

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

   14. lift-sin.f64 N/A

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

   15. lower-/.f64 N/A

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

3. Applied rewrites 99.6%

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

Alternative 2: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.97:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;\sin ky \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{1}{ky} \cdot \mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.97)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_2 -0.2)
       (*
        (sin ky)
        (/
         (* (fma (* th th) -0.16666666666666666 1.0) th)
         (hypot (sin kx) (sin ky))))
       (/ (* 1.0 (sin th)) (* (/ 1.0 ky) (hypot (sin kx) ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.97) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_2 <= -0.2) {
		tmp = sin(ky) * ((fma((th * th), -0.16666666666666666, 1.0) * th) / hypot(sin(kx), sin(ky)));
	} else {
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.97)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_2 <= -0.2)
		tmp = Float64(sin(ky) * Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(Float64(1.0 / ky) * hypot(sin(kx), ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.97], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], N[(N[Sin[ky], $MachinePrecision] * N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.96999999999999997

   1. Initial program 86.0%

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

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 79.8

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

   4. Applied rewrites 79.8%

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

   if -0.96999999999999997 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 99.0%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.1

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

   6. Applied rewrites 52.1%

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

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

   1. Initial program 95.3%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sin.f64 N/A

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 63.7%

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

   13. Taylor expanded in ky around 0

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

   15. Applied rewrites 73.8%

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

Alternative 3: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.98:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.35:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) + \left(0.5 - 0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{1}{ky} \cdot \mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.98)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_2 -0.35)
       (*
        (/
         (sin ky)
         (sqrt
          (+ (- 0.5 (* (cos (+ kx kx)) 0.5)) (- 0.5 (* 0.5 (cos (+ ky ky)))))))
        th)
       (/ (* 1.0 (sin th)) (* (/ 1.0 ky) (hypot (sin kx) ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_2 <= -0.35) {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((kx + kx)) * 0.5)) + (0.5 - (0.5 * cos((ky + ky))))))) * th;
	} else {
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.98) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_1))) * Math.sin(th);
	} else if (t_2 <= -0.35) {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (Math.cos((kx + kx)) * 0.5)) + (0.5 - (0.5 * Math.cos((ky + ky))))))) * th;
	} else {
		tmp = (1.0 * Math.sin(th)) / ((1.0 / ky) * Math.hypot(Math.sin(kx), ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	tmp = 0
	if t_2 <= -0.98:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_1))) * math.sin(th)
	elif t_2 <= -0.35:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (math.cos((kx + kx)) * 0.5)) + (0.5 - (0.5 * math.cos((ky + ky))))))) * th
	else:
		tmp = (1.0 * math.sin(th)) / ((1.0 / ky) * math.hypot(math.sin(kx), ky))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.98)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_2 <= -0.35)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)) + Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky))))))) * th);
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(Float64(1.0 / ky) * hypot(sin(kx), ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	tmp = 0.0;
	if (t_2 <= -0.98)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	elseif (t_2 <= -0.35)
		tmp = (sin(ky) / sqrt(((0.5 - (cos((kx + kx)) * 0.5)) + (0.5 - (0.5 * cos((ky + ky))))))) * th;
	else
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.98], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.35], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 85.9%

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

   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 80.5

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

   4. Applied rewrites 80.5%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.34999999999999998

   1. Initial program 99.0%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 53.4%

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

      1. lift-pow.f64 N/A

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

      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

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

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

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 53.3

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. sqr-sin-a N/A

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

      15. lower--.f64 N/A

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

      16. cos-2 N/A

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

      17. cos-sum N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. lower-+.f64 53.2

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

   6. Applied rewrites 53.2%

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

   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)))))

   1. Initial program 95.4%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sin.f64 N/A

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 62.9%

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

   13. Taylor expanded in ky around 0

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

   15. Applied rewrites 72.9%

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

Alternative 4: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{elif}\;t\_1 \leq -0.35:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) + \left(0.5 - 0.5 \cdot \cos \left(ky + ky\right)\right)}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{1}{ky} \cdot \mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.98)
     (/ (* (sin th) (sin ky)) (hypot kx (sin ky)))
     (if (<= t_1 -0.35)
       (*
        (/
         (sin ky)
         (sqrt
          (+ (- 0.5 (* (cos (+ kx kx)) 0.5)) (- 0.5 (* 0.5 (cos (+ ky ky)))))))
        th)
       (/ (* 1.0 (sin th)) (* (/ 1.0 ky) (hypot (sin kx) ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	} else if (t_1 <= -0.35) {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((kx + kx)) * 0.5)) + (0.5 - (0.5 * cos((ky + ky))))))) * th;
	} else {
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = (Math.sin(th) * Math.sin(ky)) / Math.hypot(kx, Math.sin(ky));
	} else if (t_1 <= -0.35) {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (Math.cos((kx + kx)) * 0.5)) + (0.5 - (0.5 * Math.cos((ky + ky))))))) * th;
	} else {
		tmp = (1.0 * Math.sin(th)) / ((1.0 / ky) * Math.hypot(Math.sin(kx), ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.98:
		tmp = (math.sin(th) * math.sin(ky)) / math.hypot(kx, math.sin(ky))
	elif t_1 <= -0.35:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (math.cos((kx + kx)) * 0.5)) + (0.5 - (0.5 * math.cos((ky + ky))))))) * th
	else:
		tmp = (1.0 * math.sin(th)) / ((1.0 / ky) * math.hypot(math.sin(kx), ky))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(Float64(sin(th) * sin(ky)) / hypot(kx, sin(ky)));
	elseif (t_1 <= -0.35)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)) + Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky))))))) * th);
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(Float64(1.0 / ky) * hypot(sin(kx), ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.98)
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	elseif (t_1 <= -0.35)
		tmp = (sin(ky) / sqrt(((0.5 - (cos((kx + kx)) * 0.5)) + (0.5 - (0.5 * cos((ky + ky))))))) * th;
	else
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.35], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 85.9%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 87.6

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

   8. Applied rewrites 87.6%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.34999999999999998

   1. Initial program 99.0%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 53.4%

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

      1. lift-pow.f64 N/A

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

      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

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

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

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 53.3

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. sqr-sin-a N/A

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

      15. lower--.f64 N/A

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

      16. cos-2 N/A

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

      17. cos-sum N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. lower-+.f64 53.2

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

   6. Applied rewrites 53.2%

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

   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)))))

   1. Initial program 95.4%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sin.f64 N/A

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 62.9%

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

   13. Taylor expanded in ky around 0

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

   15. Applied rewrites 72.9%

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

Alternative 5: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{elif}\;t\_1 \leq -0.35:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{1}{ky} \cdot \mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.98)
     (/ (* (sin th) (sin ky)) (hypot kx (sin ky)))
     (if (<= t_1 -0.35)
       (* (/ (sin ky) (hypot (sin kx) (sin ky))) th)
       (/ (* 1.0 (sin th)) (* (/ 1.0 ky) (hypot (sin kx) ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	} else if (t_1 <= -0.35) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	} else {
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = (Math.sin(th) * Math.sin(ky)) / Math.hypot(kx, Math.sin(ky));
	} else if (t_1 <= -0.35) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(kx), Math.sin(ky))) * th;
	} else {
		tmp = (1.0 * Math.sin(th)) / ((1.0 / ky) * Math.hypot(Math.sin(kx), ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.98:
		tmp = (math.sin(th) * math.sin(ky)) / math.hypot(kx, math.sin(ky))
	elif t_1 <= -0.35:
		tmp = (math.sin(ky) / math.hypot(math.sin(kx), math.sin(ky))) * th
	else:
		tmp = (1.0 * math.sin(th)) / ((1.0 / ky) * math.hypot(math.sin(kx), ky))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(Float64(sin(th) * sin(ky)) / hypot(kx, sin(ky)));
	elseif (t_1 <= -0.35)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * th);
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(Float64(1.0 / ky) * hypot(sin(kx), ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.98)
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	elseif (t_1 <= -0.35)
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * th;
	else
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.35], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 85.9%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 87.6

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

   8. Applied rewrites 87.6%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.34999999999999998

   1. Initial program 99.0%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 53.4%

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

   6. Applied rewrites 53.8%

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

   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)))))

   1. Initial program 95.4%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sin.f64 N/A

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 62.9%

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

   13. Taylor expanded in ky around 0

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

   15. Applied rewrites 72.9%

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

Alternative 6: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{elif}\;t\_1 \leq -0.35:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{1}{ky} \cdot \mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.98)
     (/ (* (sin th) (sin ky)) (hypot kx (sin ky)))
     (if (<= t_1 -0.35)
       (* (sin ky) (/ th (hypot (sin kx) (sin ky))))
       (/ (* 1.0 (sin th)) (* (/ 1.0 ky) (hypot (sin kx) ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	} else if (t_1 <= -0.35) {
		tmp = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	} else {
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = (Math.sin(th) * Math.sin(ky)) / Math.hypot(kx, Math.sin(ky));
	} else if (t_1 <= -0.35) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(kx), Math.sin(ky)));
	} else {
		tmp = (1.0 * Math.sin(th)) / ((1.0 / ky) * Math.hypot(Math.sin(kx), ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.98:
		tmp = (math.sin(th) * math.sin(ky)) / math.hypot(kx, math.sin(ky))
	elif t_1 <= -0.35:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(kx), math.sin(ky)))
	else:
		tmp = (1.0 * math.sin(th)) / ((1.0 / ky) * math.hypot(math.sin(kx), ky))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(Float64(sin(th) * sin(ky)) / hypot(kx, sin(ky)));
	elseif (t_1 <= -0.35)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(Float64(1.0 / ky) * hypot(sin(kx), ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.98)
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	elseif (t_1 <= -0.35)
		tmp = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	else
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.35], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 85.9%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 87.6

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

   8. Applied rewrites 87.6%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.34999999999999998

   1. Initial program 99.0%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.3%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. unpow1 N/A

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

      3. exp-to-pow N/A

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

      4. pow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. exp-to-pow N/A

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

      8. unpow1 N/A

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

      9. pow2 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-hypot.f64 53.7

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

   6. Applied rewrites 53.7%

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

   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)))))

   1. Initial program 95.4%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sin.f64 N/A

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 62.9%

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

   13. Taylor expanded in ky around 0

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

   15. Applied rewrites 72.9%

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

Alternative 7: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{elif}\;t\_1 \leq -0.35:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{1}{ky} \cdot \mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.98)
     (/ (* (sin th) (sin ky)) (hypot kx (sin ky)))
     (if (<= t_1 -0.35)
       (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))
       (/ (* 1.0 (sin th)) (* (/ 1.0 ky) (hypot (sin kx) ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	} else if (t_1 <= -0.35) {
		tmp = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	} else {
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = (Math.sin(th) * Math.sin(ky)) / Math.hypot(kx, Math.sin(ky));
	} else if (t_1 <= -0.35) {
		tmp = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	} else {
		tmp = (1.0 * Math.sin(th)) / ((1.0 / ky) * Math.hypot(Math.sin(kx), ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.98:
		tmp = (math.sin(th) * math.sin(ky)) / math.hypot(kx, math.sin(ky))
	elif t_1 <= -0.35:
		tmp = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	else:
		tmp = (1.0 * math.sin(th)) / ((1.0 / ky) * math.hypot(math.sin(kx), ky))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(Float64(sin(th) * sin(ky)) / hypot(kx, sin(ky)));
	elseif (t_1 <= -0.35)
		tmp = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)));
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(Float64(1.0 / ky) * hypot(sin(kx), ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.98)
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	elseif (t_1 <= -0.35)
		tmp = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	else
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.35], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 85.9%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 87.6

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

   8. Applied rewrites 87.6%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.34999999999999998

   1. Initial program 99.0%

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

   2. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. *-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. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 53.5

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

   4. Applied rewrites 53.5%

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

   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)))))

   1. Initial program 95.4%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sin.f64 N/A

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 62.9%

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

   13. Taylor expanded in ky around 0

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

   15. Applied rewrites 72.9%

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

Alternative 8: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.97:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.35:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{1}{ky} \cdot \mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

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 -0.97)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th))
     (if (<= t_1 -0.35)
       (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))
       (/ (* 1.0 (sin th)) (* (/ 1.0 ky) (hypot (sin kx) ky)))))))

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 <= -0.97) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	} else if (t_1 <= -0.35) {
		tmp = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	} else {
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.97) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((ky + ky)))))) * Math.sin(th);
	} else if (t_1 <= -0.35) {
		tmp = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	} else {
		tmp = (1.0 * Math.sin(th)) / ((1.0 / ky) * Math.hypot(Math.sin(kx), ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.97:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((ky + ky)))))) * math.sin(th)
	elif t_1 <= -0.35:
		tmp = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	else:
		tmp = (1.0 * math.sin(th)) / ((1.0 / ky) * math.hypot(math.sin(kx), ky))
	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 <= -0.97)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	elseif (t_1 <= -0.35)
		tmp = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)));
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(Float64(1.0 / ky) * hypot(sin(kx), ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.97)
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	elseif (t_1 <= -0.35)
		tmp = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	else
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.97], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.35], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.96999999999999997

   1. Initial program 86.0%

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

   2. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   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. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 3.7

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

   4. Applied rewrites 3.7%

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

   5. Taylor expanded in kx around 0

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

      1. pow2 N/A

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

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. cos-2 N/A

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

      5. cos-sum N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 59.3

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

   7. Applied rewrites 59.3%

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

   if -0.96999999999999997 < (/.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

   1. Initial program 99.0%

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

   2. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. *-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. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 53.0

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

   4. Applied rewrites 53.0%

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

   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)))))

   1. Initial program 95.4%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sin.f64 N/A

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 62.9%

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

   13. Taylor expanded in ky around 0

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

   15. Applied rewrites 72.9%

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

Alternative 9: 68.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{1}{ky} \cdot \mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.005)
   (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th))
   (/ (* 1.0 (sin th)) (* (/ 1.0 ky) (hypot (sin kx) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.005) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	} else {
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.005) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((ky + ky)))))) * Math.sin(th);
	} else {
		tmp = (1.0 * Math.sin(th)) / ((1.0 / ky) * Math.hypot(Math.sin(kx), ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.005:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((ky + ky)))))) * math.sin(th)
	else:
		tmp = (1.0 * math.sin(th)) / ((1.0 / ky) * math.hypot(math.sin(kx), ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.005)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(Float64(1.0 / ky) * hypot(sin(kx), ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.005)
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	else
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.005], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.0050000000000000001

   1. Initial program 99.6%

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

   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. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 12.0

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

   4. Applied rewrites 12.0%

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

   5. Taylor expanded in kx around 0

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

      1. pow2 N/A

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

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. cos-2 N/A

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

      5. cos-sum N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 59.2

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

   7. Applied rewrites 59.2%

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

   if -0.0050000000000000001 < (sin.f64 ky)

   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. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sin.f64 N/A

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 68.2%

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

   13. Taylor expanded in ky around 0

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

   15. Applied rewrites 77.4%

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

Alternative 10: 61.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.35:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{1}{ky} \cdot \mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) -0.35)
   (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) th)
   (/ (* 1.0 (sin th)) (* (/ 1.0 ky) (hypot (sin kx) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= -0.35) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	} else {
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= -0.35) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((ky + ky)))))) * th;
	} else {
		tmp = (1.0 * Math.sin(th)) / ((1.0 / ky) * Math.hypot(Math.sin(kx), ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= -0.35:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((ky + ky)))))) * th
	else:
		tmp = (1.0 * math.sin(th)) / ((1.0 / ky) * math.hypot(math.sin(kx), ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.35)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * th);
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(Float64(1.0 / ky) * hypot(sin(kx), ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.35)
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	else
		tmp = (1.0 * sin(th)) / ((1.0 / ky) * hypot(sin(kx), ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.35], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $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))))) < -0.34999999999999998

   1. Initial program 90.0%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 30.3%

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

   8. Taylor expanded in kx around 0

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

      1. pow2 N/A

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

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. cos-2 N/A

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

      5. cos-sum N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 25.2

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

   10. Applied rewrites 25.2%

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

   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)))))

   1. Initial program 95.4%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. inv-pow N/A

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

      9. lower-/.f64 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-sin.f64 N/A

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 62.9%

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

   13. Taylor expanded in ky around 0

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

   15. Applied rewrites 72.9%

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

Alternative 11: 58.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.35:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) -0.35)
   (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) th)
   (* ky (/ (sin th) (hypot (sin kx) ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= -0.35) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	} else {
		tmp = ky * (sin(th) / hypot(sin(kx), ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= -0.35) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((ky + ky)))))) * th;
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(Math.sin(kx), ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= -0.35:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((ky + ky)))))) * th
	else:
		tmp = ky * (math.sin(th) / math.hypot(math.sin(kx), ky))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.35)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * th);
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(sin(kx), ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.35)
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	else
		tmp = ky * (sin(th) / hypot(sin(kx), ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.35], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $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))))) < -0.34999999999999998

   1. Initial program 90.0%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 30.3%

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

   8. Taylor expanded in kx around 0

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

      1. pow2 N/A

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

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. cos-2 N/A

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

      5. cos-sum N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 25.2

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

   10. Applied rewrites 25.2%

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

   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)))))

   1. Initial program 95.4%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 62.9%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 70.9%

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

Alternative 12: 56.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-210}:\\ \;\;\;\;\frac{ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, ky\right)}\\ \end{array} \end{array} \]

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 -0.2)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) th)
     (if (<= t_1 2e-210)
       (* (/ ky (sqrt (- 0.5 (* 0.5 (cos (* 2.0 kx)))))) (sin th))
       (if (<= t_1 5e-9)
         (* (/ ky (sin kx)) (sin th))
         (* ky (/ (sin th) (hypot kx ky))))))))

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 <= -0.2) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	} else if (t_1 <= 2e-210) {
		tmp = (ky / sqrt((0.5 - (0.5 * cos((2.0 * kx)))))) * sin(th);
	} else if (t_1 <= 5e-9) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = ky * (sin(th) / hypot(kx, ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((ky + ky)))))) * th;
	} else if (t_1 <= 2e-210) {
		tmp = (ky / Math.sqrt((0.5 - (0.5 * Math.cos((2.0 * kx)))))) * Math.sin(th);
	} else if (t_1 <= 5e-9) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(kx, ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.2:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((ky + ky)))))) * th
	elif t_1 <= 2e-210:
		tmp = (ky / math.sqrt((0.5 - (0.5 * math.cos((2.0 * kx)))))) * math.sin(th)
	elif t_1 <= 5e-9:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = ky * (math.sin(th) / math.hypot(kx, ky))
	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 <= -0.2)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * th);
	elseif (t_1 <= 2e-210)
		tmp = Float64(Float64(ky / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx)))))) * sin(th));
	elseif (t_1 <= 5e-9)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(kx, ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	elseif (t_1 <= 2e-210)
		tmp = (ky / sqrt((0.5 - (0.5 * cos((2.0 * kx)))))) * sin(th);
	elseif (t_1 <= 5e-9)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = ky * (sin(th) / hypot(kx, ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 2e-210], N[(N[(ky / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-9], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $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))))) < -0.20000000000000001

   1. Initial program 90.2%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 47.1%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in kx around 0

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

      1. pow2 N/A

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

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. cos-2 N/A

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

      5. cos-sum N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-+.f64 24.8

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

   10. Applied rewrites 24.8%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.0000000000000001e-210

   1. Initial program 99.3%

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

   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. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 73.3

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 72.2%

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

   if 2.0000000000000001e-210 < (/.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))))) < 5.0000000000000001e-9

   1. Initial program 99.3%

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

   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. sqrt-pow1 N/A

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

      3. metadata-eval N/A

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

      4. unpow1 N/A

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

      5. lift-sin.f64 59.1

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

   4. Applied rewrites 59.1%

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

   if 5.0000000000000001e-9 < (/.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 90.9%

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

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

          \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 64.2%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.9%

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   10. Taylor expanded in ky around 0

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 42.1%

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 55.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, ky\right)}\\ \end{array} \end{array} \]

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 -0.2)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) th)
     (if (<= t_1 2e-210)
       (/ (* (sin th) ky) (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5))))
       (if (<= t_1 5e-9)
         (* (/ ky (sin kx)) (sin th))
         (* ky (/ (sin th) (hypot kx ky))))))))

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 <= -0.2) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	} else if (t_1 <= 2e-210) {
		tmp = (sin(th) * ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)));
	} else if (t_1 <= 5e-9) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = ky * (sin(th) / hypot(kx, ky));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((ky + ky)))))) * th;
	} else if (t_1 <= 2e-210) {
		tmp = (Math.sin(th) * ky) / Math.sqrt((0.5 - (Math.cos((kx + kx)) * 0.5)));
	} else if (t_1 <= 5e-9) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(kx, ky));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.2:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((ky + ky)))))) * th
	elif t_1 <= 2e-210:
		tmp = (math.sin(th) * ky) / math.sqrt((0.5 - (math.cos((kx + kx)) * 0.5)))
	elif t_1 <= 5e-9:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = ky * (math.sin(th) / math.hypot(kx, ky))
	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 <= -0.2)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * th);
	elseif (t_1 <= 2e-210)
		tmp = Float64(Float64(sin(th) * ky) / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5))));
	elseif (t_1 <= 5e-9)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(kx, ky)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	elseif (t_1 <= 2e-210)
		tmp = (sin(th) * ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)));
	elseif (t_1 <= 5e-9)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = ky * (sin(th) / hypot(kx, ky));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 2e-210], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-9], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $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))))) < -0.20000000000000001

   1. Initial program 90.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]
   3. Step-by-step derivation

   4. Applied rewrites 47.1%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]
   6. Step-by-step derivation

   7. Applied rewrites 29.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot th \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot th \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]

      4. cos-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \left(\cos ky \cdot \cos ky - \color{blue}{\sin ky \cdot \sin ky}\right)}} \cdot th \]

      5. cos-sum N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(ky + ky\right)}}} \cdot th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot th \]

      8. lower-+.f64 24.8

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th \]

   10. Applied rewrites 24.8%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}} \cdot th \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.0000000000000001e-210

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. 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. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      8. lift-sin.f64 61.5

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 61.5%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]
   5. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      2. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{1}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\left(\frac{2}{\color{blue}{2}}\right)}} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\sin kx}^{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \]

      13. lower-+.f64 72.1

          \[\leadsto \frac{\sin th \cdot ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \]

   6. Applied rewrites 72.1%

      \[\leadsto \frac{\sin th \cdot ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \]

   if 2.0000000000000001e-210 < (/.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))))) < 5.0000000000000001e-9

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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. sqrt-pow1 N/A

         \[\leadsto \frac{ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \frac{ky}{{\sin kx}^{1}} \cdot \sin th \]

      4. unpow1 N/A

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

      5. lift-sin.f64 59.1

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 5.0000000000000001e-9 < (/.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 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      14. lift-sin.f64 N/A

          \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 64.2%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.9%

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   10. Taylor expanded in ky around 0

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 42.1%

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 55.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.0005:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 0.0005)
   (* ky (/ (sin th) (hypot kx ky)))
   (/ (* (sin th) ky) (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 0.0005) {
		tmp = ky * (sin(th) / hypot(kx, ky));
	} else {
		tmp = (sin(th) * ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(kx), 2.0) <= 0.0005) {
		tmp = ky * (Math.sin(th) / Math.hypot(kx, ky));
	} else {
		tmp = (Math.sin(th) * ky) / Math.sqrt((0.5 - (Math.cos((kx + kx)) * 0.5)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(kx), 2.0) <= 0.0005:
		tmp = ky * (math.sin(th) / math.hypot(kx, ky))
	else:
		tmp = (math.sin(th) * ky) / math.sqrt((0.5 - (math.cos((kx + kx)) * 0.5)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 0.0005)
		tmp = Float64(ky * Float64(sin(th) / hypot(kx, ky)));
	else
		tmp = Float64(Float64(sin(th) * ky) / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(kx) ^ 2.0) <= 0.0005)
		tmp = ky * (sin(th) / hypot(kx, ky));
	else
		tmp = (sin(th) * ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 0.0005], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.0000000000000001e-4

   1. Initial program 88.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      14. lift-sin.f64 N/A

          \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 98.2%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.4%

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   10. Taylor expanded in ky around 0

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 70.1%

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]

   if 5.0000000000000001e-4 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   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

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. 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. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      8. lift-sin.f64 34.6

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 34.6%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]
   5. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      2. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{1}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\left(\frac{2}{\color{blue}{2}}\right)}} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\sin kx}^{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      7. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \]

      13. lower-+.f64 51.6

          \[\leadsto \frac{\sin th \cdot ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \]

   6. Applied rewrites 51.6%

      \[\leadsto \frac{\sin th \cdot ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 46.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.9:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, t\_1\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.9)
     (* t_1 (/ (sin th) (hypot kx t_1)))
     (if (<= t_2 5e-9)
       (* (/ ky (sin kx)) (sin th))
       (* ky (/ (sin th) (hypot kx ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.9) {
		tmp = t_1 * (sin(th) / hypot(kx, t_1));
	} else if (t_2 <= 5e-9) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = ky * (sin(th) / hypot(kx, ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.9)
		tmp = Float64(t_1 * Float64(sin(th) / hypot(kx, t_1)));
	elseif (t_2 <= 5e-9)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(kx, ky)));
	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]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.9], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-9], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.900000000000000022

   1. Initial program 86.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      14. lift-sin.f64 N/A

          \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 89.5%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)\right)} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

      7. lower-*.f64 36.5

         \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   9. Applied rewrites 36.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   10. Taylor expanded in ky around 0

       \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}\right)} \]

       3. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky\right)} \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky\right)} \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky\right)} \]

       6. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right)} \]

       7. lower-*.f64 42.2

          \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)} \]

   12. Applied rewrites 42.2%

       \[\leadsto \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}\right)} \]

   if -0.900000000000000022 < (/.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))))) < 5.0000000000000001e-9

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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. sqrt-pow1 N/A

         \[\leadsto \frac{ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \frac{ky}{{\sin kx}^{1}} \cdot \sin th \]

      4. unpow1 N/A

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

      5. lift-sin.f64 49.6

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 5.0000000000000001e-9 < (/.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 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      14. lift-sin.f64 N/A

          \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 64.2%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 26.9%

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   10. Taylor expanded in ky around 0

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 42.1%

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 46.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;ky \cdot \frac{th}{e^{\log \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) \cdot 0.5}}\\ \mathbf{elif}\;\sin kx \leq 0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.05)
   (* ky (/ th (exp (* (log (- 0.5 (* (cos (+ kx kx)) 0.5))) 0.5))))
   (if (<= (sin kx) 0.02)
     (* ky (/ (sin th) (hypot kx ky)))
     (* (/ ky (sin kx)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.05) {
		tmp = ky * (th / exp((log((0.5 - (cos((kx + kx)) * 0.5))) * 0.5)));
	} else if (sin(kx) <= 0.02) {
		tmp = ky * (sin(th) / hypot(kx, ky));
	} else {
		tmp = (ky / sin(kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.05) {
		tmp = ky * (th / Math.exp((Math.log((0.5 - (Math.cos((kx + kx)) * 0.5))) * 0.5)));
	} else if (Math.sin(kx) <= 0.02) {
		tmp = ky * (Math.sin(th) / Math.hypot(kx, ky));
	} else {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.05:
		tmp = ky * (th / math.exp((math.log((0.5 - (math.cos((kx + kx)) * 0.5))) * 0.5)))
	elif math.sin(kx) <= 0.02:
		tmp = ky * (math.sin(th) / math.hypot(kx, ky))
	else:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.05)
		tmp = Float64(ky * Float64(th / exp(Float64(log(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5))) * 0.5))));
	elseif (sin(kx) <= 0.02)
		tmp = Float64(ky * Float64(sin(th) / hypot(kx, ky)));
	else
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.05)
		tmp = ky * (th / exp((log((0.5 - (cos((kx + kx)) * 0.5))) * 0.5)));
	elseif (sin(kx) <= 0.02)
		tmp = ky * (sin(th) / hypot(kx, ky));
	else
		tmp = (ky / sin(kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.05], N[(ky * N[(th / N[Exp[N[(N[Log[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.02], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.050000000000000003

   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

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. 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. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      8. lift-sin.f64 15.7

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      3. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      4. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{2}\right)}} \]

      5. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      7. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{\color{blue}{2}}\right)}} \]

      8. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      9. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

      10. lift-sin.f64 15.5

          \[\leadsto ky \cdot \frac{th}{\sin kx} \]

   7. Applied rewrites 15.5%

      \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

      2. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      3. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(2 \cdot \frac{1}{2}\right)}} \]

      4. pow-pow N/A

         \[\leadsto ky \cdot \frac{th}{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}} \]

      5. pow-to-exp N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      6. lower-exp.f64 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      8. lower-log.f64 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      9. pow2 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left(\sin kx \cdot \sin kx\right) \cdot \frac{1}{2}}} \]

      10. sqr-sin-a-rev N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) \cdot \frac{1}{2}}} \]

      11. count-2-rev N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

      12. *-commutative N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      13. lower--.f64 N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      15. lift-+.f64 N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      16. lift-*.f64 26.9

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) \cdot 0.5}} \]

   9. Applied rewrites 26.9%

      \[\leadsto ky \cdot \frac{th}{e^{\log \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) \cdot 0.5}} \]

   if -0.050000000000000003 < (sin.f64 kx) < 0.0200000000000000004

   1. Initial program 88.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      14. lift-sin.f64 N/A

          \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 97.7%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.1%

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   10. Taylor expanded in ky around 0

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 69.7%

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]

   if 0.0200000000000000004 < (sin.f64 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. sqrt-pow1 N/A

         \[\leadsto \frac{ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \frac{ky}{{\sin kx}^{1}} \cdot \sin th \]

      4. unpow1 N/A

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

      5. lift-sin.f64 53.2

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 45.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.05:\\ \;\;\;\;ky \cdot \frac{th}{e^{\log \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) \cdot 0.5}}\\ \mathbf{elif}\;\sin kx \leq 0.02:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.05)
   (* ky (/ th (exp (* (log (- 0.5 (* (cos (+ kx kx)) 0.5))) 0.5))))
   (if (<= (sin kx) 0.02)
     (* ky (/ (sin th) (hypot kx ky)))
     (* ky (/ (sin th) (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.05) {
		tmp = ky * (th / exp((log((0.5 - (cos((kx + kx)) * 0.5))) * 0.5)));
	} else if (sin(kx) <= 0.02) {
		tmp = ky * (sin(th) / hypot(kx, ky));
	} else {
		tmp = ky * (sin(th) / sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.05) {
		tmp = ky * (th / Math.exp((Math.log((0.5 - (Math.cos((kx + kx)) * 0.5))) * 0.5)));
	} else if (Math.sin(kx) <= 0.02) {
		tmp = ky * (Math.sin(th) / Math.hypot(kx, ky));
	} else {
		tmp = ky * (Math.sin(th) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.05:
		tmp = ky * (th / math.exp((math.log((0.5 - (math.cos((kx + kx)) * 0.5))) * 0.5)))
	elif math.sin(kx) <= 0.02:
		tmp = ky * (math.sin(th) / math.hypot(kx, ky))
	else:
		tmp = ky * (math.sin(th) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.05)
		tmp = Float64(ky * Float64(th / exp(Float64(log(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5))) * 0.5))));
	elseif (sin(kx) <= 0.02)
		tmp = Float64(ky * Float64(sin(th) / hypot(kx, ky)));
	else
		tmp = Float64(ky * Float64(sin(th) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.05)
		tmp = ky * (th / exp((log((0.5 - (cos((kx + kx)) * 0.5))) * 0.5)));
	elseif (sin(kx) <= 0.02)
		tmp = ky * (sin(th) / hypot(kx, ky));
	else
		tmp = ky * (sin(th) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.05], N[(ky * N[(th / N[Exp[N[(N[Log[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.02], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.050000000000000003

   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

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. 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. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      8. lift-sin.f64 15.7

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 15.7%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      3. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      4. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{2}\right)}} \]

      5. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      7. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{\color{blue}{2}}\right)}} \]

      8. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      9. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

      10. lift-sin.f64 15.5

          \[\leadsto ky \cdot \frac{th}{\sin kx} \]

   7. Applied rewrites 15.5%

      \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

      2. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      3. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(2 \cdot \frac{1}{2}\right)}} \]

      4. pow-pow N/A

         \[\leadsto ky \cdot \frac{th}{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}} \]

      5. pow-to-exp N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      6. lower-exp.f64 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      8. lower-log.f64 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      9. pow2 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left(\sin kx \cdot \sin kx\right) \cdot \frac{1}{2}}} \]

      10. sqr-sin-a-rev N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) \cdot \frac{1}{2}}} \]

      11. count-2-rev N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

      12. *-commutative N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      13. lower--.f64 N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      15. lift-+.f64 N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      16. lift-*.f64 26.9

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) \cdot 0.5}} \]

   9. Applied rewrites 26.9%

      \[\leadsto ky \cdot \frac{th}{e^{\log \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) \cdot 0.5}} \]

   if -0.050000000000000003 < (sin.f64 kx) < 0.0200000000000000004

   1. Initial program 88.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      14. lift-sin.f64 N/A

          \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 97.7%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.1%

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   10. Taylor expanded in ky around 0

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 69.7%

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]

   if 0.0200000000000000004 < (sin.f64 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

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. 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. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      8. lift-sin.f64 53.2

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 53.2%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\color{blue}{\sin kx}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin \color{blue}{kx}} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      5. *-commutative N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sin \color{blue}{kx}} \]

      6. unpow1 N/A

         \[\leadsto \frac{ky \cdot \sin th}{{\sin kx}^{\color{blue}{1}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{ky \cdot \sin th}{{\sin kx}^{\left(\frac{2}{\color{blue}{2}}\right)}} \]

      8. sqrt-pow1 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      9. associate-/l* N/A

         \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2}}}} \]

      10. lower-*.f64 N/A

          \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2}}}} \]

      11. lower-/.f64 N/A

          \[\leadsto ky \cdot \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      12. lift-sin.f64 N/A

          \[\leadsto ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      13. sqrt-pow1 N/A

          \[\leadsto ky \cdot \frac{\sin th}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      14. metadata-eval N/A

          \[\leadsto ky \cdot \frac{\sin th}{{\sin kx}^{1}} \]

      15. unpow1 N/A

          \[\leadsto ky \cdot \frac{\sin th}{\sin kx} \]

      16. lift-sin.f64 53.2

          \[\leadsto ky \cdot \frac{\sin th}{\sin kx} \]

   6. Applied rewrites 53.2%

      \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 45.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.02 \cdot 10^{+30}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{e^{\log \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.02e+30)
   (* ky (/ (sin th) (hypot kx ky)))
   (* ky (/ th (exp (* (log (- 0.5 (* (cos (+ kx kx)) 0.5))) 0.5))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.02e+30) {
		tmp = ky * (sin(th) / hypot(kx, ky));
	} else {
		tmp = ky * (th / exp((log((0.5 - (cos((kx + kx)) * 0.5))) * 0.5)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.02e+30) {
		tmp = ky * (Math.sin(th) / Math.hypot(kx, ky));
	} else {
		tmp = ky * (th / Math.exp((Math.log((0.5 - (Math.cos((kx + kx)) * 0.5))) * 0.5)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.02e+30:
		tmp = ky * (math.sin(th) / math.hypot(kx, ky))
	else:
		tmp = ky * (th / math.exp((math.log((0.5 - (math.cos((kx + kx)) * 0.5))) * 0.5)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.02e+30)
		tmp = Float64(ky * Float64(sin(th) / hypot(kx, ky)));
	else
		tmp = Float64(ky * Float64(th / exp(Float64(log(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5))) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.02e+30)
		tmp = ky * (sin(th) / hypot(kx, ky));
	else
		tmp = ky * (th / exp((log((0.5 - (cos((kx + kx)) * 0.5))) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.02e+30], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(th / N[Exp[N[(N[Log[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.02e30

   1. Initial program 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      14. lift-sin.f64 N/A

          \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 70.2%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 39.4%

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   10. Taylor expanded in ky around 0

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.1%

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]

   if 1.02e30 < 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

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. 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. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      8. lift-sin.f64 34.4

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      3. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      4. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{2}\right)}} \]

      5. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      7. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{\color{blue}{2}}\right)}} \]

      8. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      9. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

      10. lift-sin.f64 22.3

          \[\leadsto ky \cdot \frac{th}{\sin kx} \]

   7. Applied rewrites 22.3%

      \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

      2. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      3. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(2 \cdot \frac{1}{2}\right)}} \]

      4. pow-pow N/A

         \[\leadsto ky \cdot \frac{th}{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}} \]

      5. pow-to-exp N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      6. lower-exp.f64 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      8. lower-log.f64 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left({\sin kx}^{2}\right) \cdot \frac{1}{2}}} \]

      9. pow2 N/A

         \[\leadsto ky \cdot \frac{th}{e^{\log \left(\sin kx \cdot \sin kx\right) \cdot \frac{1}{2}}} \]

      10. sqr-sin-a-rev N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) \cdot \frac{1}{2}}} \]

      11. count-2-rev N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right) \cdot \frac{1}{2}}} \]

      12. *-commutative N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      13. lower--.f64 N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      14. lift-cos.f64 N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      15. lift-+.f64 N/A

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}}} \]

      16. lift-*.f64 28.1

          \[\leadsto ky \cdot \frac{th}{e^{\log \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) \cdot 0.5}} \]

   9. Applied rewrites 28.1%

      \[\leadsto ky \cdot \frac{th}{e^{\log \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 45.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.02 \cdot 10^{+30}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.02e+30)
   (* ky (/ (sin th) (hypot kx ky)))
   (* ky (/ th (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.02e+30) {
		tmp = ky * (sin(th) / hypot(kx, ky));
	} else {
		tmp = ky * (th / sqrt((0.5 - (cos((kx + kx)) * 0.5))));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.02e+30) {
		tmp = ky * (Math.sin(th) / Math.hypot(kx, ky));
	} else {
		tmp = ky * (th / Math.sqrt((0.5 - (Math.cos((kx + kx)) * 0.5))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.02e+30:
		tmp = ky * (math.sin(th) / math.hypot(kx, ky))
	else:
		tmp = ky * (th / math.sqrt((0.5 - (math.cos((kx + kx)) * 0.5))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.02e+30)
		tmp = Float64(ky * Float64(sin(th) / hypot(kx, ky)));
	else
		tmp = Float64(ky * Float64(th / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.02e+30)
		tmp = ky * (sin(th) / hypot(kx, ky));
	else
		tmp = ky * (th / sqrt((0.5 - (cos((kx + kx)) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.02e+30], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(th / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.02e30

   1. Initial program 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      14. lift-sin.f64 N/A

          \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 70.2%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{kx}, \sin ky\right)} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 39.4%

      \[\leadsto \color{blue}{ky} \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   10. Taylor expanded in ky around 0

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.1%

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]

   if 1.02e30 < 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

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. 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. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      8. lift-sin.f64 34.4

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      3. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      4. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{2}\right)}} \]

      5. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      7. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{\color{blue}{2}}\right)}} \]

      8. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      9. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

      10. lift-sin.f64 22.3

          \[\leadsto ky \cdot \frac{th}{\sin kx} \]

   7. Applied rewrites 22.3%

      \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

      2. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      3. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{2}\right)}} \]

      4. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      6. pow2 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{\sin kx \cdot \sin kx}} \]

      7. sqr-sin-a-rev N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \]

      8. count-2-rev N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \]

      9. *-commutative N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \]

      10. lower--.f64 N/A

          \[\leadsto ky \cdot \frac{th}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \]

      11. lift-cos.f64 N/A

          \[\leadsto ky \cdot \frac{th}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \]

      12. lift-+.f64 N/A

          \[\leadsto ky \cdot \frac{th}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \]

      13. lift-*.f64 28.1

          \[\leadsto ky \cdot \frac{th}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \]

   9. Applied rewrites 28.1%

      \[\leadsto ky \cdot \frac{th}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 26.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 3.9 \cdot 10^{+30}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 3.9e+30) (/ (* ky th) (hypot kx ky)) (/ (* (sin th) ky) kx)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 3.9e+30) {
		tmp = (ky * th) / hypot(kx, ky);
	} else {
		tmp = (sin(th) * ky) / kx;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 3.9e+30) {
		tmp = (ky * th) / Math.hypot(kx, ky);
	} else {
		tmp = (Math.sin(th) * ky) / kx;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 3.9e+30:
		tmp = (ky * th) / math.hypot(kx, ky)
	else:
		tmp = (math.sin(th) * ky) / kx
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 3.9e+30)
		tmp = Float64(Float64(ky * th) / hypot(kx, ky));
	else
		tmp = Float64(Float64(sin(th) * ky) / kx);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 3.9e+30)
		tmp = (ky * th) / hypot(kx, ky);
	else
		tmp = (sin(th) * ky) / kx;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 3.9e+30], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / kx), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 3.90000000000000011e30

   1. Initial program 93.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]
   3. Step-by-step derivation

   4. Applied rewrites 61.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]
   6. Step-by-step derivation

   7. Applied rewrites 39.3%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot th \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot th}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot th}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot th}}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \]

      7. lift-sin.f64 36.7

         \[\leadsto \frac{\color{blue}{\sin ky} \cdot th}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot th}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \]

      14. pow2 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \]

      15. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \]

   9. Applied rewrites 38.3%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot th}{\mathsf{hypot}\left(kx, \sin ky\right)}} \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{\color{blue}{ky} \cdot th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 22.4%

       \[\leadsto \frac{\color{blue}{ky} \cdot th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   13. Taylor expanded in ky around 0

       \[\leadsto \frac{ky \cdot th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
   14. Step-by-step derivation

   15. Applied rewrites 30.9%

       \[\leadsto \frac{ky \cdot th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]

   if 3.90000000000000011e30 < th

   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

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. 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. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      8. lift-sin.f64 22.5

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 22.5%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th \cdot ky}{kx} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.4%

      \[\leadsto \frac{\sin th \cdot ky}{kx} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 24.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(kx, ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.05)
   (* ky (/ th (sin kx)))
   (/ (* ky th) (hypot kx ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.05) {
		tmp = ky * (th / sin(kx));
	} else {
		tmp = (ky * th) / hypot(kx, ky);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.05) {
		tmp = ky * (th / Math.sin(kx));
	} else {
		tmp = (ky * th) / Math.hypot(kx, ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.05:
		tmp = ky * (th / math.sin(kx))
	else:
		tmp = (ky * th) / math.hypot(kx, ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.05)
		tmp = Float64(ky * Float64(th / sin(kx)));
	else
		tmp = Float64(Float64(ky * th) / hypot(kx, ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.05)
		tmp = ky * (th / sin(kx));
	else
		tmp = (ky * th) / hypot(kx, ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.05], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $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))))) < 0.050000000000000003

   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

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. 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. sqrt-pow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      8. lift-sin.f64 34.1

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 34.1%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      3. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      4. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{2}\right)}} \]

      5. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      7. sqrt-pow1 N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{\color{blue}{2}}\right)}} \]

      8. metadata-eval N/A

         \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

      9. unpow1 N/A

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

      10. lift-sin.f64 21.9

          \[\leadsto ky \cdot \frac{th}{\sin kx} \]

   7. Applied rewrites 21.9%

      \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]

   if 0.050000000000000003 < (/.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 90.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]
   3. Step-by-step derivation

   4. Applied rewrites 46.4%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]
   6. Step-by-step derivation

   7. Applied rewrites 30.1%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot th \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot th}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot th}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot th}}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \]

      7. lift-sin.f64 28.6

         \[\leadsto \frac{\color{blue}{\sin ky} \cdot th}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky \cdot th}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \]

      13. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \]

      14. pow2 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \]

      15. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \]

   9. Applied rewrites 30.6%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot th}{\mathsf{hypot}\left(kx, \sin ky\right)}} \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{\color{blue}{ky} \cdot th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 10.8%

       \[\leadsto \frac{\color{blue}{ky} \cdot th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

   13. Taylor expanded in ky around 0

       \[\leadsto \frac{ky \cdot th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
   14. Step-by-step derivation

   15. Applied rewrites 20.8%

       \[\leadsto \frac{ky \cdot th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 21.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \frac{ky \cdot th}{\mathsf{hypot}\left(kx, ky\right)} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (/ (* ky th) (hypot kx ky)))

C

double code(double kx, double ky, double th) {
	return (ky * th) / hypot(kx, ky);
}

Java

public static double code(double kx, double ky, double th) {
	return (ky * th) / Math.hypot(kx, ky);
}

Python

def code(kx, ky, th):
	return (ky * th) / math.hypot(kx, ky)

Julia

function code(kx, ky, th)
	return Float64(Float64(ky * th) / hypot(kx, ky))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (ky * th) / hypot(kx, ky);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(ky * th), $MachinePrecision] / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in th around 0

   \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]
3. Step-by-step derivation

4. Applied rewrites 48.3%

   \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]
6. Step-by-step derivation

7. Applied rewrites 31.3%

   \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot th} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot th \]

   3. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin ky}}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \cdot th \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sin ky \cdot th}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin ky \cdot th}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin ky \cdot th}}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \]

   7. lift-sin.f64 29.3

      \[\leadsto \frac{\color{blue}{\sin ky} \cdot th}{\sqrt{{kx}^{2} + {\sin ky}^{2}}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sin ky \cdot th}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \]

   12. lift-pow.f64 N/A

       \[\leadsto \frac{\sin ky \cdot th}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \]

   13. lift-sin.f64 N/A

       \[\leadsto \frac{\sin ky \cdot th}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \]

   14. pow2 N/A

       \[\leadsto \frac{\sin ky \cdot th}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \]

   15. lower-hypot.f64 N/A

       \[\leadsto \frac{\sin ky \cdot th}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \]

9. Applied rewrites 30.5%

   \[\leadsto \color{blue}{\frac{\sin ky \cdot th}{\mathsf{hypot}\left(kx, \sin ky\right)}} \]

10. Taylor expanded in ky around 0

    \[\leadsto \frac{\color{blue}{ky} \cdot th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]
11. Step-by-step derivation

12. Applied rewrites 18.1%

    \[\leadsto \frac{\color{blue}{ky} \cdot th}{\mathsf{hypot}\left(kx, \sin ky\right)} \]

13. Taylor expanded in ky around 0

    \[\leadsto \frac{ky \cdot th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
14. Step-by-step derivation

15. Applied rewrites 24.8%

    \[\leadsto \frac{ky \cdot th}{\mathsf{hypot}\left(kx, \color{blue}{ky}\right)} \]
16. Add Preprocessing

Alternative 23: 13.7% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ ky \cdot \frac{th}{kx} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* ky (/ th kx)))

C

double code(double kx, double ky, double th) {
	return ky * (th / kx);
}

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 * (th / kx)
end function

Java

public static double code(double kx, double ky, double th) {
	return ky * (th / kx);
}

Python

def code(kx, ky, th):
	return ky * (th / kx)

Julia

function code(kx, ky, th)
	return Float64(ky * Float64(th / kx))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = ky * (th / kx);
end

Wolfram

code[kx_, ky_, th_] := N[(ky * N[(th / kx), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in ky around 0

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
3. 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. sqrt-pow1 N/A

      \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

   7. unpow1 N/A

      \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   8. lift-sin.f64 24.9

      \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

4. Applied rewrites 24.9%

   \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

5. Taylor expanded in th around 0

   \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

   2. lower-*.f64 N/A

      \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

   3. unpow1 N/A

      \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

   4. metadata-eval N/A

      \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{2}\right)}} \]

   5. sqrt-pow1 N/A

      \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

   6. lower-/.f64 N/A

      \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

   7. sqrt-pow1 N/A

      \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{\color{blue}{2}}\right)}} \]

   8. metadata-eval N/A

      \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

   9. unpow1 N/A

      \[\leadsto ky \cdot \frac{th}{\sin kx} \]

   10. lift-sin.f64 16.2

       \[\leadsto ky \cdot \frac{th}{\sin kx} \]

7. Applied rewrites 16.2%

   \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]

8. Taylor expanded in kx around 0

   \[\leadsto ky \cdot \frac{th}{kx} \]
9. Step-by-step derivation

   1. lower-/.f64 13.7

      \[\leadsto ky \cdot \frac{th}{kx} \]

10. Applied rewrites 13.7%

    \[\leadsto ky \cdot \frac{th}{kx} \]
11. Add Preprocessing

Alternative 24: 13.7% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ th \cdot \frac{ky}{kx} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* th (/ ky kx)))

C

double code(double kx, double ky, double th) {
	return th * (ky / kx);
}

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 = th * (ky / kx)
end function

Java

public static double code(double kx, double ky, double th) {
	return th * (ky / kx);
}

Python

def code(kx, ky, th):
	return th * (ky / kx)

Julia

function code(kx, ky, th)
	return Float64(th * Float64(ky / kx))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th * (ky / kx);
end

Wolfram

code[kx_, ky_, th_] := N[(th * N[(ky / kx), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in ky around 0

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
3. 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. sqrt-pow1 N/A

      \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\sin th \cdot ky}{{\sin kx}^{1}} \]

   7. unpow1 N/A

      \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   8. lift-sin.f64 24.9

      \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

4. Applied rewrites 24.9%

   \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

5. Taylor expanded in th around 0

   \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

   2. lower-*.f64 N/A

      \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

   3. unpow1 N/A

      \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

   4. metadata-eval N/A

      \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{2}\right)}} \]

   5. sqrt-pow1 N/A

      \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

   6. lower-/.f64 N/A

      \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

   7. sqrt-pow1 N/A

      \[\leadsto ky \cdot \frac{th}{{\sin kx}^{\left(\frac{2}{\color{blue}{2}}\right)}} \]

   8. metadata-eval N/A

      \[\leadsto ky \cdot \frac{th}{{\sin kx}^{1}} \]

   9. unpow1 N/A

      \[\leadsto ky \cdot \frac{th}{\sin kx} \]

   10. lift-sin.f64 16.2

       \[\leadsto ky \cdot \frac{th}{\sin kx} \]

7. Applied rewrites 16.2%

   \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]

8. Taylor expanded in kx around 0

   \[\leadsto \frac{ky \cdot th}{kx} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{ky \cdot th}{kx} \]

   2. *-commutative N/A

      \[\leadsto \frac{th \cdot ky}{kx} \]

   3. lower-*.f64 12.7

      \[\leadsto \frac{th \cdot ky}{kx} \]

10. Applied rewrites 12.7%

    \[\leadsto \frac{th \cdot ky}{kx} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{th \cdot ky}{kx} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{th \cdot ky}{kx} \]

    3. associate-/l* N/A

       \[\leadsto th \cdot \frac{ky}{kx} \]

    4. lower-*.f64 N/A

       \[\leadsto th \cdot \frac{ky}{kx} \]

    5. lower-/.f64 13.7

       \[\leadsto th \cdot \frac{ky}{kx} \]

12. Applied rewrites 13.7%

    \[\leadsto th \cdot \frac{ky}{kx} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025133 
(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)))