Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.9% → 99.7%

Time: 7.6s

Alternatives: 25

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: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.7

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

3. Applied rewrites 99.7%

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

Alternative 2: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(0.008333333333333333 \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\left(\frac{1}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.98:\\ \;\;\;\;\left(e^{\log t\_1 \cdot -1} \cdot \sin ky\right) \cdot \left(th \cdot \left(1 - 0.16666666666666666 \cdot \left(th \cdot th\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.999)
     (*
      (/
       (sin ky)
       (hypot
        (sin ky)
        (*
         (fma
          (- (* 0.008333333333333333 (* kx kx)) 0.16666666666666666)
          (* kx kx)
          1.0)
         kx)))
      (sin th))
     (if (<= t_2 -0.2)
       (/ (* (sin ky) th) t_1)
       (if (<= t_2 0.2)
         (* (* (/ 1.0 (hypot (sin kx) ky)) (sin ky)) (sin th))
         (if (<= t_2 0.98)
           (*
            (* (exp (* (log t_1) -1.0)) (sin ky))
            (* th (- 1.0 (* 0.16666666666666666 (* th th)))))
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(((0.008333333333333333 * (kx * kx)) - 0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	} else if (t_2 <= -0.2) {
		tmp = (sin(ky) * th) / t_1;
	} else if (t_2 <= 0.2) {
		tmp = ((1.0 / hypot(sin(kx), ky)) * sin(ky)) * sin(th);
	} else if (t_2 <= 0.98) {
		tmp = (exp((log(t_1) * -1.0)) * sin(ky)) * (th * (1.0 - (0.16666666666666666 * (th * th))));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.999)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(Float64(Float64(0.008333333333333333 * Float64(kx * kx)) - 0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	elseif (t_2 <= -0.2)
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(Float64(1.0 / hypot(sin(kx), ky)) * sin(ky)) * sin(th));
	elseif (t_2 <= 0.98)
		tmp = Float64(Float64(exp(Float64(log(t_1) * -1.0)) * sin(ky)) * Float64(th * Float64(1.0 - Float64(0.16666666666666666 * Float64(th * th)))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(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, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(N[(N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[(N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.98], N[(N[(N[Exp[N[(N[Log[t$95$1], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(th * N[(1.0 - N[(0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999

   1. Initial program 86.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

   if -0.998999999999999999 < (/.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 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-to-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 48.0%

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

   8. Applied rewrites 48.1%

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

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

   1. Initial program 99.2%

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

   2. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 94.0%

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

   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))))) < 0.97999999999999998

   1. Initial program 99.3%

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

   2. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lower-hypot.f64 N/A

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

      5. inv-pow N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. pow-to-exp N/A

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

      9. lower-exp.f64 N/A

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.6

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

   9. Applied rewrites 52.6%

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

   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)))))

   1. Initial program 86.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 95.6%

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

Alternative 3: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(0.008333333333333333 \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\left(\frac{1}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.98:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin kx) (sin ky))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.999)
     (*
      (/
       (sin ky)
       (hypot
        (sin ky)
        (*
         (fma
          (- (* 0.008333333333333333 (* kx kx)) 0.16666666666666666)
          (* kx kx)
          1.0)
         kx)))
      (sin th))
     (if (<= t_2 -0.2)
       t_1
       (if (<= t_2 0.2)
         (* (* (/ 1.0 (hypot (sin kx) ky)) (sin ky)) (sin th))
         (if (<= t_2 0.98)
           t_1
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(((0.008333333333333333 * (kx * kx)) - 0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	} else if (t_2 <= -0.2) {
		tmp = t_1;
	} else if (t_2 <= 0.2) {
		tmp = ((1.0 / hypot(sin(kx), ky)) * sin(ky)) * sin(th);
	} else if (t_2 <= 0.98) {
		tmp = t_1;
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.999)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(Float64(Float64(0.008333333333333333 * Float64(kx * kx)) - 0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	elseif (t_2 <= -0.2)
		tmp = t_1;
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(Float64(1.0 / hypot(sin(kx), ky)) * sin(ky)) * sin(th));
	elseif (t_2 <= 0.98)
		tmp = t_1;
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(N[(N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], t$95$1, If[LessEqual[t$95$2, 0.2], N[(N[(N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.98], t$95$1, N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999

   1. Initial program 86.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.97999999999999998

   1. Initial program 99.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-to-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.2%

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

   8. Applied rewrites 50.4%

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

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

   1. Initial program 99.2%

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

   2. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 94.0%

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

   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)))))

   1. Initial program 86.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 95.6%

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

Alternative 4: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;\left(\frac{1}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.98:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))))
   (if (<= t_1 -0.999)
     (*
      (/
       (sin ky)
       (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
      (sin th))
     (if (<= t_1 -0.2)
       t_2
       (if (<= t_1 0.2)
         (* (* (/ 1.0 (hypot (sin kx) ky)) (sin ky)) (sin th))
         (if (<= t_1 0.98)
           t_2
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 0.2) {
		tmp = ((1.0 / hypot(sin(kx), ky)) * sin(ky)) * sin(th);
	} else if (t_1 <= 0.98) {
		tmp = t_2;
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 0.2)
		tmp = Float64(Float64(Float64(1.0 / hypot(sin(kx), ky)) * sin(ky)) * sin(th));
	elseif (t_1 <= 0.98)
		tmp = t_2;
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 0.2], N[(N[(N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.98], t$95$2, N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999

   1. Initial program 86.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 98.5

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

   6. Applied rewrites 98.5%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.97999999999999998

   1. Initial program 99.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-to-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.2%

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

   8. Applied rewrites 50.4%

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

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

   1. Initial program 99.2%

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

   2. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 94.0%

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

   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)))))

   1. Initial program 86.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 95.6%

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

Alternative 5: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_2 \leq -0.999:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\left(\frac{1}{\mathsf{hypot}\left(\sin kx, ky\right)} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.98:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))))
   (if (<= t_2 -0.999)
     t_1
     (if (<= t_2 -0.2)
       t_3
       (if (<= t_2 0.2)
         (* (* (/ 1.0 (hypot (sin kx) ky)) (sin ky)) (sin th))
         (if (<= t_2 0.98) t_3 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = t_1;
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 0.2) {
		tmp = ((1.0 / hypot(sin(kx), ky)) * sin(ky)) * sin(th);
	} else if (t_2 <= 0.98) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_3 = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = t_1;
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 0.2) {
		tmp = ((1.0 / Math.hypot(Math.sin(kx), ky)) * Math.sin(ky)) * Math.sin(th);
	} else if (t_2 <= 0.98) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_3 = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if t_2 <= -0.999:
		tmp = t_1
	elif t_2 <= -0.2:
		tmp = t_3
	elif t_2 <= 0.2:
		tmp = ((1.0 / math.hypot(math.sin(kx), ky)) * math.sin(ky)) * math.sin(th)
	elif t_2 <= 0.98:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	tmp = 0.0
	if (t_2 <= -0.999)
		tmp = t_1;
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(Float64(1.0 / hypot(sin(kx), ky)) * sin(ky)) * sin(th));
	elseif (t_2 <= 0.98)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_3 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (t_2 <= -0.999)
		tmp = t_1;
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 0.2)
		tmp = ((1.0 / hypot(sin(kx), ky)) * sin(ky)) * sin(th);
	elseif (t_2 <= 0.98)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999], t$95$1, If[LessEqual[t$95$2, -0.2], t$95$3, If[LessEqual[t$95$2, 0.2], N[(N[(N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.98], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999 or 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)))))

   1. Initial program 86.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 97.0%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.97999999999999998

   1. Initial program 99.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-to-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.2%

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

   8. Applied rewrites 50.4%

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

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

   1. Initial program 99.2%

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

   2. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 94.0%

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

Alternative 6: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_2 \leq -0.999:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.98:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))))
   (if (<= t_2 -0.999)
     t_1
     (if (<= t_2 -0.2)
       t_3
       (if (<= t_2 0.2)
         (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
         (if (<= t_2 0.98) t_3 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = t_1;
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 0.2) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else if (t_2 <= 0.98) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_3 = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = t_1;
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 0.2) {
		tmp = (Math.sin(ky) / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	} else if (t_2 <= 0.98) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_3 = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if t_2 <= -0.999:
		tmp = t_1
	elif t_2 <= -0.2:
		tmp = t_3
	elif t_2 <= 0.2:
		tmp = (math.sin(ky) / math.hypot(ky, math.sin(kx))) * math.sin(th)
	elif t_2 <= 0.98:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	tmp = 0.0
	if (t_2 <= -0.999)
		tmp = t_1;
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(sin(ky) / hypot(ky, sin(kx))) * sin(th));
	elseif (t_2 <= 0.98)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_3 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (t_2 <= -0.999)
		tmp = t_1;
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 0.2)
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	elseif (t_2 <= 0.98)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999], t$95$1, If[LessEqual[t$95$2, -0.2], t$95$3, If[LessEqual[t$95$2, 0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.98], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999 or 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)))))

   1. Initial program 86.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 97.0%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.97999999999999998

   1. Initial program 99.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-to-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.2%

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

   8. Applied rewrites 50.4%

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

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

   1. Initial program 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 94.1%

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

Alternative 7: 84.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.98:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))))
   (if (<= t_1 -0.999)
     (/ (* (sin th) (sin ky)) (hypot kx (sin ky)))
     (if (<= t_1 -0.2)
       t_2
       (if (<= t_1 0.2)
         (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
         (if (<= t_1 0.98)
           t_2
           (if (<= t_1 1.0)
             (sin th)
             (*
              (/
               ky
               (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
              (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 0.2) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else if (t_1 <= 0.98) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_2 = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (Math.sin(th) * Math.sin(ky)) / Math.hypot(kx, Math.sin(ky));
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 0.2) {
		tmp = (Math.sin(ky) / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	} else if (t_1 <= 0.98) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_2 = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if t_1 <= -0.999:
		tmp = (math.sin(th) * math.sin(ky)) / math.hypot(kx, math.sin(ky))
	elif t_1 <= -0.2:
		tmp = t_2
	elif t_1 <= 0.2:
		tmp = (math.sin(ky) / math.hypot(ky, math.sin(kx))) * math.sin(th)
	elif t_1 <= 0.98:
		tmp = t_2
	elif t_1 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(th) * sin(ky)) / hypot(kx, sin(ky)));
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 0.2)
		tmp = Float64(Float64(sin(ky) / hypot(ky, sin(kx))) * sin(th));
	elseif (t_1 <= 0.98)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_2 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (t_1 <= -0.999)
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 0.2)
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	elseif (t_1 <= 0.98)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.98], t$95$2, If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999

   1. Initial program 86.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 91.4

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

      11. lift-hypot.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

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

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

      15. +-commutative N/A

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

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

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

      17. lower-hypot.f64 N/A

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

   8. Applied rewrites 91.4%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.97999999999999998

   1. Initial program 99.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-to-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 98.8%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.2%

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

   8. Applied rewrites 50.4%

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

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

   1. Initial program 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 94.1%

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

   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))))) < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 95.6

         \[\leadsto \sin th \]

   4. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\sin th} \]

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

   1. Initial program 4.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

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

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

       3. lower--.f64 N/A

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

       4. metadata-eval N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 99.7

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

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.98:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))))
   (if (<= t_1 -0.999)
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) (sin th))
     (if (<= t_1 -0.2)
       t_2
       (if (<= t_1 2e-14)
         (* (* (/ 1.0 (sin kx)) (sin ky)) (sin th))
         (if (<= t_1 0.98)
           t_2
           (if (<= t_1 1.0)
             (sin th)
             (*
              (/
               ky
               (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
              (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * sin(th);
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 2e-14) {
		tmp = ((1.0 / sin(kx)) * sin(ky)) * sin(th);
	} else if (t_1 <= 0.98) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_2 = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((ky + ky)) * 0.5)))) * Math.sin(th);
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 2e-14) {
		tmp = ((1.0 / Math.sin(kx)) * Math.sin(ky)) * Math.sin(th);
	} else if (t_1 <= 0.98) {
		tmp = t_2;
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_2 = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if t_1 <= -0.999:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((ky + ky)) * 0.5)))) * math.sin(th)
	elif t_1 <= -0.2:
		tmp = t_2
	elif t_1 <= 2e-14:
		tmp = ((1.0 / math.sin(kx)) * math.sin(ky)) * math.sin(th)
	elif t_1 <= 0.98:
		tmp = t_2
	elif t_1 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * sin(th));
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(Float64(1.0 / sin(kx)) * sin(ky)) * sin(th));
	elseif (t_1 <= 0.98)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_2 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (t_1 <= -0.999)
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * sin(th);
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 2e-14)
		tmp = ((1.0 / sin(kx)) * sin(ky)) * sin(th);
	elseif (t_1 <= 0.98)
		tmp = t_2;
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 2e-14], N[(N[(N[(1.0 / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.98], t$95$2, If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999

   1. Initial program 86.1%

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

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) + {\sin ky}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot {kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right), \color{blue}{{kx}^{2}}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{kx}}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) \cdot {kx}^{2} + 1, {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, {kx}^{2}, 1\right), {\color{blue}{kx}}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, {kx}^{2}, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, {kx}^{2}, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, {kx}^{2}, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, {kx}^{2}, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      15. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      16. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      18. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      19. lower-*.f64 62.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(kx \cdot kx\right) - 0.3333333333333333, kx \cdot kx, 1\right), kx \cdot kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 62.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(kx \cdot kx\right) - 0.3333333333333333, kx \cdot kx, 1\right), kx \cdot kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 62.1

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

   7. Applied rewrites 62.1%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 2e-14 < (/.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 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-to-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 49.7%

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

   8. Applied rewrites 50.4%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-14

   1. Initial program 99.2%

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

   2. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. lift-sin.f64 62.3

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

   7. Applied rewrites 62.3%

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

   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))))) < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 95.6

         \[\leadsto \sin th \]

   4. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\sin th} \]

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

   1. Initial program 4.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

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

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

       3. lower--.f64 N/A

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

       4. metadata-eval N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 99.7

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

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.17:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.17)
   (/ (* (sin th) (sin ky)) (hypot kx (sin ky)))
   (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.17) {
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	} else {
		tmp = (sin(ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.17) {
		tmp = (Math.sin(th) * Math.sin(ky)) / Math.hypot(kx, Math.sin(ky));
	} else {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((kx + kx)) * 0.5)))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 0.17:
		tmp = (math.sin(th) * math.sin(ky)) / math.hypot(kx, math.sin(ky))
	else:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((kx + kx)) * 0.5)))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.17)
		tmp = Float64(Float64(sin(th) * sin(ky)) / hypot(kx, sin(ky)));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 0.17)
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	else
		tmp = (sin(ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.17], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.170000000000000012

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 72.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 67.3

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

      11. lift-hypot.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

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

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

      15. +-commutative N/A

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

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

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

      17. lower-hypot.f64 N/A

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

   8. Applied rewrites 67.3%

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

   if 0.170000000000000012 < kx

   1. Initial program 99.4%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 17.6

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

   4. Applied rewrites 17.6%

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

   5. Taylor expanded in ky around 0

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

      1. pow2 N/A

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

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

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

      3. pow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

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

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 60.0

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

   7. Applied rewrites 60.0%

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

Alternative 10: 61.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.35:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.72:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.35)
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) (sin th))
     (if (<= t_1 2e-94)
       (* (* (/ 1.0 (sin kx)) (sin ky)) (sin th))
       (if (<= t_1 0.72)
         (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) (sin th))
         (if (<= t_1 1.0)
           (sin th)
           (*
            (/ ky (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.35) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * sin(th);
	} else if (t_1 <= 2e-94) {
		tmp = ((1.0 / sin(kx)) * sin(ky)) * sin(th);
	} else if (t_1 <= 0.72) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * sin(th);
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.35) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((ky + ky)) * 0.5)))) * Math.sin(th);
	} else if (t_1 <= 2e-94) {
		tmp = ((1.0 / Math.sin(kx)) * Math.sin(ky)) * Math.sin(th);
	} else if (t_1 <= 0.72) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((kx + kx)) * 0.5)))) * Math.sin(th);
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.35:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((ky + ky)) * 0.5)))) * math.sin(th)
	elif t_1 <= 2e-94:
		tmp = ((1.0 / math.sin(kx)) * math.sin(ky)) * math.sin(th)
	elif t_1 <= 0.72:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((kx + kx)) * 0.5)))) * math.sin(th)
	elif t_1 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.35)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * sin(th));
	elseif (t_1 <= 2e-94)
		tmp = Float64(Float64(Float64(1.0 / sin(kx)) * sin(ky)) * sin(th));
	elseif (t_1 <= 0.72)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * sin(th));
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.35)
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * sin(th);
	elseif (t_1 <= 2e-94)
		tmp = ((1.0 / sin(kx)) * sin(ky)) * sin(th);
	elseif (t_1 <= 0.72)
		tmp = (sin(ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * sin(th);
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.35], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-94], N[(N[(N[(1.0 / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.72], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 90.4%

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

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) + {\sin ky}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot {kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right), \color{blue}{{kx}^{2}}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{kx}}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) \cdot {kx}^{2} + 1, {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, {kx}^{2}, 1\right), {\color{blue}{kx}}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, {kx}^{2}, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, {kx}^{2}, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, {kx}^{2}, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, {kx}^{2}, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      15. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      16. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      18. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot \left(kx \cdot kx\right) - \frac{1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      19. lower-*.f64 43.6

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(kx \cdot kx\right) - 0.3333333333333333, kx \cdot kx, 1\right), kx \cdot kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 43.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(kx \cdot kx\right) - 0.3333333333333333, kx \cdot kx, 1\right), kx \cdot kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 48.7

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

   7. Applied rewrites 48.7%

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

   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))))) < 1.9999999999999999e-94

   1. Initial program 99.2%

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

   2. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. lift-sin.f64 62.5

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

   7. Applied rewrites 62.5%

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

   if 1.9999999999999999e-94 < (/.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))))) < 0.71999999999999997

   1. Initial program 99.0%

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

   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

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 19.9

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

   4. Applied rewrites 19.9%

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

   5. Taylor expanded in ky around 0

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

      1. pow2 N/A

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

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

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

      3. pow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

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

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 42.9

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

   7. Applied rewrites 42.9%

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

   if 0.71999999999999997 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 80.5

         \[\leadsto \sin th \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\sin th} \]

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

   1. Initial program 4.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

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

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

       3. lower--.f64 N/A

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

       4. metadata-eval N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 99.7

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

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 59.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot \sin th\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, t\_2\right)}} \cdot th\\ \mathbf{elif}\;t\_3 \leq 10^{-161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.72:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) (sin th)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -1.0)
     (* (/ (sin ky) (sqrt (fma kx kx t_2))) th)
     (if (<= t_3 1e-161)
       t_1
       (if (<= t_3 2e-94)
         (* (* (/ 1.0 (sin kx)) (sin ky)) (sin th))
         (if (<= t_3 0.72)
           t_1
           (if (<= t_3 1.0)
             (sin th)
             (*
              (/
               ky
               (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
              (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * sin(th);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(fma(kx, kx, t_2))) * th;
	} else if (t_3 <= 1e-161) {
		tmp = t_1;
	} else if (t_3 <= 2e-94) {
		tmp = ((1.0 / sin(kx)) * sin(ky)) * sin(th);
	} else if (t_3 <= 0.72) {
		tmp = t_1;
	} else if (t_3 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * sin(th))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(kx, kx, t_2))) * th);
	elseif (t_3 <= 1e-161)
		tmp = t_1;
	elseif (t_3 <= 2e-94)
		tmp = Float64(Float64(Float64(1.0 / sin(kx)) * sin(ky)) * sin(th));
	elseif (t_3 <= 0.72)
		tmp = t_1;
	elseif (t_3 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(kx * kx + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 1e-161], t$95$1, If[LessEqual[t$95$3, 2e-94], N[(N[(N[(1.0 / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.72], t$95$1, If[LessEqual[t$95$3, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 85.6%

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

   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

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 63.5

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

   4. Applied rewrites 63.5%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 32.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

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

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 43.7

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

   9. Applied rewrites 43.7%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000003e-161 or 1.9999999999999999e-94 < (/.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))))) < 0.71999999999999997

   1. Initial program 99.2%

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

   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

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 31.6

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

   4. Applied rewrites 31.6%

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

   5. Taylor expanded in ky around 0

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

      1. pow2 N/A

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

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

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

      3. pow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

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

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 53.2

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

   7. Applied rewrites 53.2%

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

   if 1.00000000000000003e-161 < (/.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))))) < 1.9999999999999999e-94

   1. Initial program 98.9%

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

   2. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

      1. lift-sin.f64 61.9

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

   7. Applied rewrites 61.9%

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

   if 0.71999999999999997 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.7%

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

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 80.5

         \[\leadsto \sin th \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\sin th} \]

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

   1. Initial program 4.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

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

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

       3. lower--.f64 N/A

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

       4. metadata-eval N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 99.7

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

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 12: 53.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.35:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.35)
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) th)
     (if (<= t_1 2e-94)
       (* (* (/ 1.0 (sin kx)) (sin ky)) (sin th))
       (if (<= t_1 5e-7)
         (* (* (/ 1.0 (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) ky) (sin th))
         (if (<= t_1 1.0)
           (sin th)
           (*
            (/ ky (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.35) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * th;
	} else if (t_1 <= 2e-94) {
		tmp = ((1.0 / sin(kx)) * sin(ky)) * sin(th);
	} else if (t_1 <= 5e-7) {
		tmp = ((1.0 / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * ky) * sin(th);
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.35) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((ky + ky)) * 0.5)))) * th;
	} else if (t_1 <= 2e-94) {
		tmp = ((1.0 / Math.sin(kx)) * Math.sin(ky)) * Math.sin(th);
	} else if (t_1 <= 5e-7) {
		tmp = ((1.0 / Math.sqrt((0.5 - (Math.cos((kx + kx)) * 0.5)))) * ky) * Math.sin(th);
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.35:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((ky + ky)) * 0.5)))) * th
	elif t_1 <= 2e-94:
		tmp = ((1.0 / math.sin(kx)) * math.sin(ky)) * math.sin(th)
	elif t_1 <= 5e-7:
		tmp = ((1.0 / math.sqrt((0.5 - (math.cos((kx + kx)) * 0.5)))) * ky) * math.sin(th)
	elif t_1 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.35)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * th);
	elseif (t_1 <= 2e-94)
		tmp = Float64(Float64(Float64(1.0 / sin(kx)) * sin(ky)) * sin(th));
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * ky) * sin(th));
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.35)
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * th;
	elseif (t_1 <= 2e-94)
		tmp = ((1.0 / sin(kx)) * sin(ky)) * sin(th);
	elseif (t_1 <= 5e-7)
		tmp = ((1.0 / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * ky) * sin(th);
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.35], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 2e-94], N[(N[(N[(1.0 / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(N[(N[(1.0 / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 90.4%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 43.8

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 43.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}}} \cdot th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      5. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(ky + ky\right) \cdot \frac{1}{2}}} \cdot th \]

      6. lower-+.f64 25.3

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot th \]

   10. Applied rewrites 25.3%

       \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\cos \left(ky + ky\right) \cdot 0.5}}} \cdot th \]

   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))))) < 1.9999999999999999e-94

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\sin ky}\right) \cdot \sin th \]

      3. sqrt-div N/A

         \[\leadsto \left(\frac{\sqrt{1}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin \color{blue}{ky}\right) \cdot \sin th \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin \color{blue}{ky}\right) \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin ky\right) \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \cdot \sin ky\right) \cdot \sin th \]

      8. lower-hypot.f64 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      10. lift-sin.f64 N/A

          \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      11. lift-sin.f64 99.5

          \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right)} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 62.5

         \[\leadsto \left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th \]

   7. Applied rewrites 62.5%

      \[\leadsto \left(\frac{1}{\sin kx} \cdot \sin ky\right) \cdot \sin th \]

   if 1.9999999999999999e-94 < (/.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))))) < 4.99999999999999977e-7

   1. Initial program 98.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. pow1/2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\frac{1}{2}}}} \cdot \sin th \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      9. lower-exp.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{e^{\color{blue}{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

   3. Applied rewrites 68.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right) \cdot 0.5}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \color{blue}{ky}\right) \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \color{blue}{ky}\right) \cdot \sin th \]

      3. sqrt-div N/A

         \[\leadsto \left(\frac{\sqrt{1}}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. lift-cos.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. lower-+.f64 67.9

          \[\leadsto \left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   6. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   if 4.99999999999999977e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 65.6

         \[\leadsto \sin th \]

   4. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 4.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   5. Step-by-step derivation

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       3. lower--.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       4. metadata-eval N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot {\color{blue}{ky}}^{2}\right), kx\right)} \cdot \sin th \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \color{blue}{{ky}^{2}}\right), kx\right)} \cdot \sin th \]

       6. pow2 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

       7. lower-*.f64 99.7

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 13: 53.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.35:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.35)
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) th)
     (if (<= t_1 2e-94)
       (* (/ (sin ky) (sin kx)) (sin th))
       (if (<= t_1 5e-7)
         (* (* (/ 1.0 (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) ky) (sin th))
         (if (<= t_1 1.0)
           (sin th)
           (*
            (/ ky (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.35) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * th;
	} else if (t_1 <= 2e-94) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 5e-7) {
		tmp = ((1.0 / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * ky) * sin(th);
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.35) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((ky + ky)) * 0.5)))) * th;
	} else if (t_1 <= 2e-94) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else if (t_1 <= 5e-7) {
		tmp = ((1.0 / Math.sqrt((0.5 - (Math.cos((kx + kx)) * 0.5)))) * ky) * Math.sin(th);
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.35:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((ky + ky)) * 0.5)))) * th
	elif t_1 <= 2e-94:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	elif t_1 <= 5e-7:
		tmp = ((1.0 / math.sqrt((0.5 - (math.cos((kx + kx)) * 0.5)))) * ky) * math.sin(th)
	elif t_1 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.35)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * th);
	elseif (t_1 <= 2e-94)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * ky) * sin(th));
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.35)
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * th;
	elseif (t_1 <= 2e-94)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	elseif (t_1 <= 5e-7)
		tmp = ((1.0 / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * ky) * sin(th);
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.35], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 2e-94], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(N[(N[(1.0 / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.34999999999999998

   1. Initial program 90.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 43.8

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 43.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}}} \cdot th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      5. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(ky + ky\right) \cdot \frac{1}{2}}} \cdot th \]

      6. lower-+.f64 25.3

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot th \]

   10. Applied rewrites 25.3%

       \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\cos \left(ky + ky\right) \cdot 0.5}}} \cdot th \]

   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))))) < 1.9999999999999999e-94

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lift-sin.f64 62.5

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 62.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 1.9999999999999999e-94 < (/.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))))) < 4.99999999999999977e-7

   1. Initial program 98.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. pow1/2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\frac{1}{2}}}} \cdot \sin th \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      9. lower-exp.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{e^{\color{blue}{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

   3. Applied rewrites 68.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right) \cdot 0.5}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \color{blue}{ky}\right) \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \color{blue}{ky}\right) \cdot \sin th \]

      3. sqrt-div N/A

         \[\leadsto \left(\frac{\sqrt{1}}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. lift-cos.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. lower-+.f64 67.9

          \[\leadsto \left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   6. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   if 4.99999999999999977e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 65.6

         \[\leadsto \sin th \]

   4. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 4.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   5. Step-by-step derivation

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       3. lower--.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       4. metadata-eval N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot {\color{blue}{ky}}^{2}\right), kx\right)} \cdot \sin th \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \color{blue}{{ky}^{2}}\right), kx\right)} \cdot \sin th \]

       6. pow2 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

       7. lower-*.f64 99.7

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 14: 53.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) th)
     (if (<= t_1 2e-94)
       (* (/ ky (sin kx)) (sin th))
       (if (<= t_1 5e-7)
         (* (* (/ 1.0 (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) ky) (sin th))
         (if (<= t_1 1.0)
           (sin th)
           (*
            (/ ky (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * th;
	} else if (t_1 <= 2e-94) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_1 <= 5e-7) {
		tmp = ((1.0 / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * ky) * sin(th);
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((ky + ky)) * 0.5)))) * th;
	} else if (t_1 <= 2e-94) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else if (t_1 <= 5e-7) {
		tmp = ((1.0 / Math.sqrt((0.5 - (Math.cos((kx + kx)) * 0.5)))) * ky) * Math.sin(th);
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.2:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((ky + ky)) * 0.5)))) * th
	elif t_1 <= 2e-94:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	elif t_1 <= 5e-7:
		tmp = ((1.0 / math.sqrt((0.5 - (math.cos((kx + kx)) * 0.5)))) * ky) * math.sin(th)
	elif t_1 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * th);
	elseif (t_1 <= 2e-94)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * ky) * sin(th));
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * th;
	elseif (t_1 <= 2e-94)
		tmp = (ky / sin(kx)) * sin(th);
	elseif (t_1 <= 5e-7)
		tmp = ((1.0 / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * ky) * sin(th);
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 2e-94], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(N[(N[(1.0 / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 90.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 42.7

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 42.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 22.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}}} \cdot th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      5. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(ky + ky\right) \cdot \frac{1}{2}}} \cdot th \]

      6. lower-+.f64 24.8

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot th \]

   10. Applied rewrites 24.8%

       \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\cos \left(ky + ky\right) \cdot 0.5}}} \cdot th \]

   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))))) < 1.9999999999999999e-94

   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

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 63.7

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.9999999999999999e-94 < (/.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))))) < 4.99999999999999977e-7

   1. Initial program 98.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. pow1/2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\frac{1}{2}}}} \cdot \sin th \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      9. lower-exp.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{e^{\color{blue}{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

   3. Applied rewrites 68.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right) \cdot 0.5}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \color{blue}{ky}\right) \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \color{blue}{ky}\right) \cdot \sin th \]

      3. sqrt-div N/A

         \[\leadsto \left(\frac{\sqrt{1}}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right) \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. lift-cos.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \left(\frac{1}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. lower-+.f64 67.9

          \[\leadsto \left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   6. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   if 4.99999999999999977e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 65.6

         \[\leadsto \sin th \]

   4. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 4.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   5. Step-by-step derivation

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       3. lower--.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       4. metadata-eval N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot {\color{blue}{ky}}^{2}\right), kx\right)} \cdot \sin th \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \color{blue}{{ky}^{2}}\right), kx\right)} \cdot \sin th \]

       6. pow2 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

       7. lower-*.f64 99.7

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 15: 53.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot \left(\sin th \cdot ky\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) th)
     (if (<= t_1 2e-94)
       (* (/ ky (sin kx)) (sin th))
       (if (<= t_1 5e-7)
         (* (/ 1.0 (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) (* (sin th) ky))
         (if (<= t_1 1.0)
           (sin th)
           (*
            (/ ky (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * th;
	} else if (t_1 <= 2e-94) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_1 <= 5e-7) {
		tmp = (1.0 / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * (sin(th) * ky);
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((ky + ky)) * 0.5)))) * th;
	} else if (t_1 <= 2e-94) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else if (t_1 <= 5e-7) {
		tmp = (1.0 / Math.sqrt((0.5 - (Math.cos((kx + kx)) * 0.5)))) * (Math.sin(th) * ky);
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.2:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((ky + ky)) * 0.5)))) * th
	elif t_1 <= 2e-94:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	elif t_1 <= 5e-7:
		tmp = (1.0 / math.sqrt((0.5 - (math.cos((kx + kx)) * 0.5)))) * (math.sin(th) * ky)
	elif t_1 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * th);
	elseif (t_1 <= 2e-94)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(1.0 / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * Float64(sin(th) * ky));
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * th;
	elseif (t_1 <= 2e-94)
		tmp = (ky / sin(kx)) * sin(th);
	elseif (t_1 <= 5e-7)
		tmp = (1.0 / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * (sin(th) * ky);
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 2e-94], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(N[(1.0 / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 90.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 42.7

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 42.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 22.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}}} \cdot th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      5. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(ky + ky\right) \cdot \frac{1}{2}}} \cdot th \]

      6. lower-+.f64 24.8

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot th \]

   10. Applied rewrites 24.8%

       \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\cos \left(ky + ky\right) \cdot 0.5}}} \cdot th \]

   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))))) < 1.9999999999999999e-94

   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

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 63.7

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.9999999999999999e-94 < (/.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))))) < 4.99999999999999977e-7

   1. Initial program 98.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. pow1/2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\frac{1}{2}}}} \cdot \sin th \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      9. lower-exp.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{e^{\color{blue}{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

   3. Applied rewrites 68.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right) \cdot 0.5}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \color{blue}{\left(ky \cdot \sin th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \color{blue}{\left(ky \cdot \sin th\right)} \]

      3. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(\color{blue}{ky} \cdot \sin th\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sin th\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(\color{blue}{ky} \cdot \sin th\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sin th\right) \]

      7. lower--.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sin th\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      11. count-2-rev N/A

          \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \cdot \left(\sin th \cdot \color{blue}{ky}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}}} \cdot \left(\sin th \cdot \color{blue}{ky}\right) \]

   6. Applied rewrites 67.7%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot \left(\sin th \cdot ky\right)} \]

   if 4.99999999999999977e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 65.6

         \[\leadsto \sin th \]

   4. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 4.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   5. Step-by-step derivation

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       3. lower--.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       4. metadata-eval N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot {\color{blue}{ky}}^{2}\right), kx\right)} \cdot \sin th \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \color{blue}{{ky}^{2}}\right), kx\right)} \cdot \sin th \]

       6. pow2 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

       7. lower-*.f64 99.7

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) th)
     (if (<= t_1 2e-7)
       (* (/ ky (sin kx)) (sin th))
       (if (<= t_1 1.0)
         (sin th)
         (*
          (/ ky (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
          (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * th;
	} else if (t_1 <= 2e-7) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((ky + ky)) * 0.5)))) * th;
	} else if (t_1 <= 2e-7) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.2:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((ky + ky)) * 0.5)))) * th
	elif t_1 <= 2e-7:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	elif t_1 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * th);
	elseif (t_1 <= 2e-7)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * th;
	elseif (t_1 <= 2e-7)
		tmp = (ky / sin(kx)) * sin(th);
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 90.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 42.7

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 42.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 22.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}}} \cdot th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot th \]

      5. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(ky + ky\right) \cdot \frac{1}{2}}} \cdot th \]

      6. lower-+.f64 24.8

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot th \]

   10. Applied rewrites 24.8%

       \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{\cos \left(ky + ky\right) \cdot 0.5}}} \cdot th \]

   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))))) < 1.9999999999999999e-7

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 61.8

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.9999999999999999e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 65.6

         \[\leadsto \sin th \]

   4. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 4.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   5. Step-by-step derivation

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       3. lower--.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       4. metadata-eval N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot {\color{blue}{ky}}^{2}\right), kx\right)} \cdot \sin th \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \color{blue}{{ky}^{2}}\right), kx\right)} \cdot \sin th \]

       6. pow2 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

       7. lower-*.f64 99.7

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 48.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 2e-7)
     (* (/ ky (sin kx)) (sin th))
     (if (<= t_1 1.0)
       (sin th)
       (*
        (/ ky (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
        (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 2e-7:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	elif t_1 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 2e-7)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 2e-7)
		tmp = (ky / sin(kx)) * sin(th);
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-7], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 35.1

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1.9999999999999999e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 65.6

         \[\leadsto \sin th \]

   4. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 4.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   5. Step-by-step derivation

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       3. lower--.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       4. metadata-eval N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot {\color{blue}{ky}}^{2}\right), kx\right)} \cdot \sin th \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \color{blue}{{ky}^{2}}\right), kx\right)} \cdot \sin th \]

       6. pow2 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

       7. lower-*.f64 99.7

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 46.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 2e-7)
     (/ (* (sin th) ky) (sin kx))
     (if (<= t_1 1.0)
       (sin th)
       (*
        (/ ky (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
        (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = (sin(th) * ky) / sin(kx);
	} else if (t_1 <= 1.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 2e-7) {
		tmp = (Math.sin(th) * ky) / Math.sin(kx);
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 2e-7:
		tmp = (math.sin(th) * ky) / math.sin(kx)
	elif t_1 <= 1.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 2e-7)
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 2e-7)
		tmp = (sin(th) * ky) / sin(kx);
	elseif (t_1 <= 1.0)
		tmp = sin(th);
	else
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-7], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.9999999999999999e-7

   1. Initial program 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sin kx}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin \color{blue}{kx}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin \color{blue}{kx}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      5. lift-sin.f64 34.0

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 34.0%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   if 1.9999999999999999e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 65.6

         \[\leadsto \sin th \]

   4. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 4.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   5. Step-by-step derivation

   6. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       3. lower--.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       4. metadata-eval N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot {\color{blue}{ky}}^{2}\right), kx\right)} \cdot \sin th \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \color{blue}{{ky}^{2}}\right), kx\right)} \cdot \sin th \]

       6. pow2 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

       7. lower-*.f64 99.7

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 45.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right), kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-27)
   (*
    (/ ky (hypot (* ky (- 1.0 (* 0.16666666666666666 (* ky ky)))) kx))
    (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-27) {
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 2e-27) {
		tmp = (ky / Math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 2e-27:
		tmp = (ky / math.hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-27)
		tmp = Float64(Float64(ky / hypot(Float64(ky * Float64(1.0 - Float64(0.16666666666666666 * Float64(ky * ky)))), kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 2e-27)
		tmp = (ky / hypot((ky * (1.0 - (0.16666666666666666 * (ky * ky)))), kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-27], N[(N[(ky / N[Sqrt[N[(ky * N[(1.0 - N[(0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 2.0000000000000001e-27

   1. Initial program 91.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   5. Step-by-step derivation

   6. Applied rewrites 61.2%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 44.2%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       3. lower--.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {ky}^{2}}\right), kx\right)} \cdot \sin th \]

       4. metadata-eval N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot {\color{blue}{ky}}^{2}\right), kx\right)} \cdot \sin th \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \color{blue}{{ky}^{2}}\right), kx\right)} \cdot \sin th \]

       6. pow2 N/A

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - \frac{1}{6} \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

       7. lower-*.f64 44.4

          \[\leadsto \frac{ky}{\mathsf{hypot}\left(ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot \color{blue}{ky}\right)\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 44.4%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 - 0.16666666666666666 \cdot \left(ky \cdot ky\right)\right)}, kx\right)} \cdot \sin th \]

   if 2.0000000000000001e-27 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 59.1

         \[\leadsto \sin th \]

   4. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 37.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, ky \cdot ky\right)}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 5e-38)
     (* (/ (sin ky) (sqrt (fma kx kx (* ky ky)))) th)
     (if (<= t_1 5e-7)
       (* (* (/ 1.0 (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) ky) th)
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 5e-38) {
		tmp = (sin(ky) / sqrt(fma(kx, kx, (ky * ky)))) * th;
	} else if (t_1 <= 5e-7) {
		tmp = ((1.0 / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * ky) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 5e-38)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(kx, kx, Float64(ky * ky)))) * th);
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * ky) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-38], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(kx * kx + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(N[(N[(1.0 / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.00000000000000033e-38

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 45.2

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 45.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 28.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, {ky}^{2}\right)}} \cdot th \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, ky \cdot ky\right)}} \cdot th \]

      2. lower-*.f64 23.3

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, ky \cdot ky\right)}} \cdot th \]

   10. Applied rewrites 23.3%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, ky \cdot ky\right)}} \cdot th \]

   if 5.00000000000000033e-38 < (/.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))))) < 4.99999999999999977e-7

   1. Initial program 98.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. pow1/2 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{\left({\sin kx}^{2} + {\sin ky}^{2}\right)}^{\frac{1}{2}}}} \cdot \sin th \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      9. lower-exp.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{e^{\color{blue}{\log \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

   3. Applied rewrites 69.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right) \cdot 0.5}}} \cdot \sin th \]

   4. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{e^{\log \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right) \cdot \frac{1}{2}}} \cdot \color{blue}{th} \]
   5. Step-by-step derivation

   6. Applied rewrites 36.4%

      \[\leadsto \frac{\sin ky}{e^{\log \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)\right) \cdot 0.5}} \cdot \color{blue}{th} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot th \]
   8. Step-by-step derivation

      1. exp-to-pow N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot th \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot th \]

      3. sqr-sin-a-rev N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot th \]

      4. pow1/2 N/A

         \[\leadsto \left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right) \cdot th \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \color{blue}{ky}\right) \cdot th \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \color{blue}{ky}\right) \cdot th \]

   9. Applied rewrites 36.2%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot ky\right)} \cdot th \]

   if 4.99999999999999977e-7 < (/.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 91.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 64.7

         \[\leadsto \sin th \]

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 37.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, ky \cdot ky\right)}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-23)
   (* (/ (sin ky) (sqrt (fma kx kx (* ky ky)))) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-23) {
		tmp = (sin(ky) / sqrt(fma(kx, kx, (ky * ky)))) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-23)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(kx, kx, Float64(ky * ky)))) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-23], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(kx * kx + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $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))))) < 3.99999999999999984e-23

   1. Initial program 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 45.1

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 45.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 28.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, {ky}^{2}\right)}} \cdot th \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, ky \cdot ky\right)}} \cdot th \]

      2. lower-*.f64 23.3

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, ky \cdot ky\right)}} \cdot th \]

   10. Applied rewrites 23.3%

       \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, ky \cdot ky\right)}} \cdot th \]

   if 3.99999999999999984e-23 < (/.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 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 63.0

         \[\leadsto \sin th \]

   4. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 34.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-23)
   (* (/ (sin ky) (sqrt (* kx kx))) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-23) {
		tmp = (sin(ky) / sqrt((kx * kx))) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-23) then
        tmp = (sin(ky) / sqrt((kx * kx))) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

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)))) <= 4e-23) {
		tmp = (Math.sin(ky) / Math.sqrt((kx * kx))) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 4e-23:
		tmp = (math.sin(ky) / math.sqrt((kx * kx))) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-23)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(kx * kx))) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-23)
		tmp = (sin(ky) / sqrt((kx * kx))) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-23], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(kx * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $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))))) < 3.99999999999999984e-23

   1. Initial program 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   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

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 45.1

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 45.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 28.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \color{blue}{th} \]

   8. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\sqrt{{kx}^{\color{blue}{2}}}} \cdot th \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx}} \cdot th \]

      2. lower-*.f64 19.5

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx}} \cdot th \]

   10. Applied rewrites 19.5%

       \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx}}} \cdot th \]

   if 3.99999999999999984e-23 < (/.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 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 63.0

         \[\leadsto \sin th \]

   4. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 31.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5.8 \cdot 10^{-25}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\left(th \cdot th\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5.8e-25)
   (* -0.16666666666666666 (* (* th th) th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5.8e-25) {
		tmp = -0.16666666666666666 * ((th * th) * th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5.8d-25) then
        tmp = (-0.16666666666666666d0) * ((th * th) * th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

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)))) <= 5.8e-25) {
		tmp = -0.16666666666666666 * ((th * th) * th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5.8e-25:
		tmp = -0.16666666666666666 * ((th * th) * th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5.8e-25)
		tmp = Float64(-0.16666666666666666 * Float64(Float64(th * th) * th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5.8e-25)
		tmp = -0.16666666666666666 * ((th * th) * th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5.8e-25], N[(-0.16666666666666666 * N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $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))))) < 5.8000000000000001e-25

   1. Initial program 95.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 3.6

         \[\leadsto \sin th \]

   4. Applied rewrites 3.6%

      \[\leadsto \color{blue}{\sin th} \]

   5. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 3.4

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   7. Applied rewrites 3.4%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   8. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{6} \cdot {th}^{3} \]

      2. unpow3 N/A

         \[\leadsto \frac{-1}{6} \cdot \left(\left(th \cdot th\right) \cdot th\right) \]

      3. pow2 N/A

         \[\leadsto \frac{-1}{6} \cdot \left({th}^{2} \cdot th\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{6} \cdot \left({th}^{2} \cdot th\right) \]

      5. pow2 N/A

         \[\leadsto \frac{-1}{6} \cdot \left(\left(th \cdot th\right) \cdot th\right) \]

      6. lift-*.f64 14.6

         \[\leadsto -0.16666666666666666 \cdot \left(\left(th \cdot th\right) \cdot th\right) \]

   10. Applied rewrites 14.6%

       \[\leadsto -0.16666666666666666 \cdot \left(\left(th \cdot th\right) \cdot \color{blue}{th}\right) \]

   if 5.8000000000000001e-25 < (/.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 91.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 62.9

         \[\leadsto \sin th \]

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 15.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 2 \cdot 10^{-308}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\left(th \cdot th\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      2e-308)
   (* -0.16666666666666666 (* (* th th) th))
   th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 2e-308) {
		tmp = -0.16666666666666666 * ((th * th) * th);
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)) <= 2d-308) then
        tmp = (-0.16666666666666666d0) * ((th * th) * th)
    else
        tmp = th
    end if
    code = tmp
end function

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)))) * Math.sin(th)) <= 2e-308) {
		tmp = -0.16666666666666666 * ((th * th) * th);
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)) <= 2e-308:
		tmp = -0.16666666666666666 * ((th * th) * th)
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-308)
		tmp = Float64(-0.16666666666666666 * Float64(Float64(th * th) * th));
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-308)
		tmp = -0.16666666666666666 * ((th * th) * th);
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[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], 2e-308], N[(-0.16666666666666666 * N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.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))))) (sin.f64 th)) < 1.9999999999999998e-308

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 23.2

         \[\leadsto \sin th \]

   4. Applied rewrites 23.2%

      \[\leadsto \color{blue}{\sin th} \]

   5. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 13.3

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   7. Applied rewrites 13.3%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   8. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{6} \cdot {th}^{3} \]

      2. unpow3 N/A

         \[\leadsto \frac{-1}{6} \cdot \left(\left(th \cdot th\right) \cdot th\right) \]

      3. pow2 N/A

         \[\leadsto \frac{-1}{6} \cdot \left({th}^{2} \cdot th\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{6} \cdot \left({th}^{2} \cdot th\right) \]

      5. pow2 N/A

         \[\leadsto \frac{-1}{6} \cdot \left(\left(th \cdot th\right) \cdot th\right) \]

      6. lift-*.f64 16.7

         \[\leadsto -0.16666666666666666 \cdot \left(\left(th \cdot th\right) \cdot th\right) \]

   10. Applied rewrites 16.7%

       \[\leadsto -0.16666666666666666 \cdot \left(\left(th \cdot th\right) \cdot \color{blue}{th}\right) \]

   if 1.9999999999999998e-308 < (*.f64 (/.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))))) (sin.f64 th))

   1. Initial program 93.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 25.5

         \[\leadsto \sin th \]

   4. Applied rewrites 25.5%

      \[\leadsto \color{blue}{\sin th} \]

   5. Taylor expanded in th around 0

      \[\leadsto th \]
   6. Step-by-step derivation

   7. Applied rewrites 14.6%

      \[\leadsto th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 14.0% accurate, 170.4× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return 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 = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 93.9%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
3. Step-by-step derivation

   1. lift-sin.f64 24.3

      \[\leadsto \sin th \]

4. Applied rewrites 24.3%

   \[\leadsto \color{blue}{\sin th} \]

5. Taylor expanded in th around 0

   \[\leadsto th \]
6. Step-by-step derivation

7. Applied rewrites 14.0%

   \[\leadsto th \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025113 
(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)))