Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 10.6s

Alternatives: 22

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: 93.8% 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 95.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 82.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := t\_1 \cdot ky\\ t_4 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_4}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(t\_1 \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(t\_3, \sin kx\right)}\\ \mathbf{elif}\;t\_2 \leq 0.97:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{t\_4} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fma (* ky ky) -0.16666666666666666 1.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (* t_1 ky))
        (t_4 (hypot (sin ky) (sin kx))))
   (if (<= t_2 -1.0)
     (* (/ 1.0 (sin ky)) (* (sin th) (fabs (sin ky))))
     (if (<= t_2 -0.05)
       (/ (* (sin ky) th) t_4)
       (if (<= t_2 2e-5)
         (/ (* (* t_1 (sin th)) ky) (hypot t_3 (sin kx)))
         (if (<= t_2 0.97)
           (* (/ (* (fma (* th th) -0.16666666666666666 1.0) th) t_4) (sin ky))
           (if (<= t_2 2.0)
             (sin th)
             (*
              (/
               (sin th)
               (hypot
                (sin ky)
                (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
              t_3))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = t_1 * ky;
	double t_4 = hypot(sin(ky), sin(kx));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (1.0 / sin(ky)) * (sin(th) * fabs(sin(ky)));
	} else if (t_2 <= -0.05) {
		tmp = (sin(ky) * th) / t_4;
	} else if (t_2 <= 2e-5) {
		tmp = ((t_1 * sin(th)) * ky) / hypot(t_3, sin(kx));
	} else if (t_2 <= 0.97) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * th) / t_4) * sin(ky);
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * t_3;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(Float64(ky * ky), -0.16666666666666666, 1.0)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(t_1 * ky)
	t_4 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(1.0 / sin(ky)) * Float64(sin(th) * abs(sin(ky))));
	elseif (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) * th) / t_4);
	elseif (t_2 <= 2e-5)
		tmp = Float64(Float64(Float64(t_1 * sin(th)) * ky) / hypot(t_3, sin(kx)));
	elseif (t_2 <= 0.97)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / t_4) * sin(ky));
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * t_3);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $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[(t$95$1 * ky), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$2, 2e-5], N[(N[(N[(t$95$1 * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[t$95$3 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.97], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / t$95$4), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 94.7%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 5.2%

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

   10. Applied rewrites 98.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000016e-5

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 93.3

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 93.3

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

   10. Applied rewrites 93.3%

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

   if 2.00000000000000016e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.96999999999999997

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

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

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 2.5

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   10. Applied rewrites 99.8%

       \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.97:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := t\_1 \cdot ky\\ t_4 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(t\_1 \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(t\_3, \sin kx\right)}\\ \mathbf{elif}\;t\_2 \leq 0.97:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fma (* ky ky) -0.16666666666666666 1.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (* t_1 ky))
        (t_4 (/ (* (sin ky) th) (hypot (sin ky) (sin kx)))))
   (if (<= t_2 -1.0)
     (* (/ 1.0 (sin ky)) (* (sin th) (fabs (sin ky))))
     (if (<= t_2 -0.05)
       t_4
       (if (<= t_2 2e-5)
         (/ (* (* t_1 (sin th)) ky) (hypot t_3 (sin kx)))
         (if (<= t_2 0.97)
           t_4
           (if (<= t_2 2.0)
             (sin th)
             (*
              (/
               (sin th)
               (hypot
                (sin ky)
                (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
              t_3))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = t_1 * ky;
	double t_4 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (1.0 / sin(ky)) * (sin(th) * fabs(sin(ky)));
	} else if (t_2 <= -0.05) {
		tmp = t_4;
	} else if (t_2 <= 2e-5) {
		tmp = ((t_1 * sin(th)) * ky) / hypot(t_3, sin(kx));
	} else if (t_2 <= 0.97) {
		tmp = t_4;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * t_3;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(Float64(ky * ky), -0.16666666666666666, 1.0)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(t_1 * ky)
	t_4 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(1.0 / sin(ky)) * Float64(sin(th) * abs(sin(ky))));
	elseif (t_2 <= -0.05)
		tmp = t_4;
	elseif (t_2 <= 2e-5)
		tmp = Float64(Float64(Float64(t_1 * sin(th)) * ky) / hypot(t_3, sin(kx)));
	elseif (t_2 <= 0.97)
		tmp = t_4;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * t_3);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $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[(t$95$1 * ky), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], t$95$4, If[LessEqual[t$95$2, 2e-5], N[(N[(N[(t$95$1 * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[t$95$3 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.97], t$95$4, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 94.7%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 5.2%

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

   10. Applied rewrites 98.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 2.00000000000000016e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.96999999999999997

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 59.4

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

   7. Applied rewrites 59.4%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000016e-5

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 93.3

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 93.3

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

   10. Applied rewrites 93.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

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

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 2.5

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.97:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \left(\left(\sin ky \cdot th\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{\left(2 - \cos \left(-2 \cdot ky\right)\right) - \cos \left(-2 \cdot kx\right)}}\\ t_3 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)\\ t_4 := t\_3 \cdot ky\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{\left(t\_3 \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(t\_4, \sin kx\right)}\\ \mathbf{elif}\;t\_1 \leq 0.97:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot t\_4\\ \end{array} \end{array} \]

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) (sqrt 2.0))
          (sqrt (/ 1.0 (- (- 2.0 (cos (* -2.0 ky))) (cos (* -2.0 kx)))))))
        (t_3 (fma (* ky ky) -0.16666666666666666 1.0))
        (t_4 (* t_3 ky)))
   (if (<= t_1 -1.0)
     (* (/ 1.0 (sin ky)) (* (sin th) (fabs (sin ky))))
     (if (<= t_1 -0.05)
       t_2
       (if (<= t_1 0.1)
         (/ (* (* t_3 (sin th)) ky) (hypot t_4 (sin kx)))
         (if (<= t_1 0.97)
           t_2
           (if (<= t_1 2.0)
             (sin th)
             (*
              (/
               (sin th)
               (hypot
                (sin ky)
                (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
              t_4))))))))

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) * sqrt(2.0)) * sqrt((1.0 / ((2.0 - cos((-2.0 * ky))) - cos((-2.0 * kx)))));
	double t_3 = fma((ky * ky), -0.16666666666666666, 1.0);
	double t_4 = t_3 * ky;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (1.0 / sin(ky)) * (sin(th) * fabs(sin(ky)));
	} else if (t_1 <= -0.05) {
		tmp = t_2;
	} else if (t_1 <= 0.1) {
		tmp = ((t_3 * sin(th)) * ky) / hypot(t_4, sin(kx));
	} else if (t_1 <= 0.97) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * t_4;
	}
	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(Float64(sin(ky) * th) * sqrt(2.0)) * sqrt(Float64(1.0 / Float64(Float64(2.0 - cos(Float64(-2.0 * ky))) - cos(Float64(-2.0 * kx))))))
	t_3 = fma(Float64(ky * ky), -0.16666666666666666, 1.0)
	t_4 = Float64(t_3 * ky)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(1.0 / sin(ky)) * Float64(sin(th) * abs(sin(ky))));
	elseif (t_1 <= -0.05)
		tmp = t_2;
	elseif (t_1 <= 0.1)
		tmp = Float64(Float64(Float64(t_3 * sin(th)) * ky) / hypot(t_4, sin(kx)));
	elseif (t_1 <= 0.97)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * t_4);
	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[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * ky), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$2, If[LessEqual[t$95$1, 0.1], N[(N[(N[(t$95$3 * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[t$95$4 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.97], t$95$2, If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 94.7%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 5.2%

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

   10. Applied rewrites 98.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.96999999999999997

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. div-add-rev N/A

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

      9. sqrt-div N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate--r+ N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 N/A

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

      14. cos-neg-rev N/A

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

      15. lower-cos.f64 N/A

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

      16. distribute-lft-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. cos-neg-rev N/A

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

      20. lower-cos.f64 N/A

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

   9. Applied rewrites 57.8%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 93.3

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 93.3

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

   10. Applied rewrites 93.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

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

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 2.5

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\left(\left(\sin ky \cdot th\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{\left(2 - \cos \left(-2 \cdot ky\right)\right) - \cos \left(-2 \cdot kx\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.97:\\ \;\;\;\;\left(\left(\sin ky \cdot th\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{\left(2 - \cos \left(-2 \cdot ky\right)\right) - \cos \left(-2 \cdot kx\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin ky\\ \mathbf{elif}\;t\_3 \leq 0.9999999999977582:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin ky) (sin kx))))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_2))) (sin th))
     (if (<= t_3 -0.05)
       t_1
       (if (<= t_3 0.05)
         (*
          (/
           (sin th)
           (hypot
            (*
             (fma
              (-
               (*
                (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
                (* ky ky))
               0.16666666666666666)
              (* ky ky)
              1.0)
             ky)
            (sin kx)))
          (sin ky))
         (if (<= t_3 0.9999999999977582)
           t_1
           (*
            (/
             (sin th)
             (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
            (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	} else if (t_3 <= -0.05) {
		tmp = t_1;
	} else if (t_3 <= 0.05) {
		tmp = (sin(th) / hypot((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky), sin(kx))) * sin(ky);
	} else if (t_3 <= 0.9999999999977582) {
		tmp = t_1;
	} else {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_2))) * sin(th));
	elseif (t_3 <= -0.05)
		tmp = t_1;
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(sin(th) / hypot(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(ky));
	elseif (t_3 <= 0.9999999999977582)
		tmp = t_1;
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky));
	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[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $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[(N[(kx * kx), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.05], t$95$1, If[LessEqual[t$95$3, 0.05], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9999999999977582], t$95$1, N[(N[(N[Sin[th], $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[ky], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 94.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.9999999999977582

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.6

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

   7. Applied rewrites 58.6%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.6

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}, \sin kx\right)} \cdot \sin ky \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot ky}, \sin kx\right)} \cdot \sin ky \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot ky}, \sin kx\right)} \cdot \sin ky \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin ky \]

   if 0.9999999999977582 < (/.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 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 82.6

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 6: 83.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky \cdot th}{t\_1}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.002:\\ \;\;\;\;\frac{\sin th}{t\_1} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \mathbf{elif}\;t\_4 \leq 0.9999999999977582:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (/ (* (sin ky) th) t_1))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_3)))))
   (if (<= t_4 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.05)
       t_2
       (if (<= t_4 0.002)
         (* (/ (sin th) t_1) (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
         (if (<= t_4 0.9999999999977582)
           t_2
           (*
            (/
             (sin th)
             (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
            (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = (sin(ky) * th) / t_1;
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.05) {
		tmp = t_2;
	} else if (t_4 <= 0.002) {
		tmp = (sin(th) / t_1) * (fma((ky * ky), -0.16666666666666666, 1.0) * ky);
	} else if (t_4 <= 0.9999999999977582) {
		tmp = t_2;
	} else {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = Float64(Float64(sin(ky) * th) / t_1)
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.05)
		tmp = t_2;
	elseif (t_4 <= 0.002)
		tmp = Float64(Float64(sin(th) / t_1) * Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky));
	elseif (t_4 <= 0.9999999999977582)
		tmp = t_2;
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.05], t$95$2, If[LessEqual[t$95$4, 0.002], N[(N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9999999999977582], t$95$2, N[(N[(N[Sin[th], $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[ky], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 94.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 2e-3 < (/.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))))) < 0.9999999999977582

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.6

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

   7. Applied rewrites 58.6%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-3

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.6

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if 0.9999999999977582 < (/.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 82.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 82.6

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 7: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_4 := \frac{\sin ky \cdot th}{t\_3}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0.002:\\ \;\;\;\;\frac{\sin th}{t\_3} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \mathbf{elif}\;t\_2 \leq 0.9999999999977582:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (/
           (sin th)
           (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
          (sin ky)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (hypot (sin ky) (sin kx)))
        (t_4 (/ (* (sin ky) th) t_3)))
   (if (<= t_2 -1.0)
     t_1
     (if (<= t_2 -0.05)
       t_4
       (if (<= t_2 0.002)
         (* (/ (sin th) t_3) (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
         (if (<= t_2 0.9999999999977582) t_4 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = hypot(sin(ky), sin(kx));
	double t_4 = (sin(ky) * th) / t_3;
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.05) {
		tmp = t_4;
	} else if (t_2 <= 0.002) {
		tmp = (sin(th) / t_3) * (fma((ky * ky), -0.16666666666666666, 1.0) * ky);
	} else if (t_2 <= 0.9999999999977582) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = hypot(sin(ky), sin(kx))
	t_4 = Float64(Float64(sin(ky) * th) / t_3)
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.05)
		tmp = t_4;
	elseif (t_2 <= 0.002)
		tmp = Float64(Float64(sin(th) / t_3) * Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky));
	elseif (t_2 <= 0.9999999999977582)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[th], $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[ky], $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[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], t$95$1, If[LessEqual[t$95$2, -0.05], t$95$4, If[LessEqual[t$95$2, 0.002], N[(N[(N[Sin[th], $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999999999977582], t$95$4, 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))))) < -1 or 0.9999999999977582 < (/.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 87.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.6

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 2e-3 < (/.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))))) < 0.9999999999977582

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.6

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

   7. Applied rewrites 58.6%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-3

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.6

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

Alternative 8: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)\\ t_4 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_4}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(t\_3 \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(t\_3 \cdot ky, \sin kx\right)}\\ \mathbf{elif}\;t\_2 \leq 0.98:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{t\_4} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (/
           (sin th)
           (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
          (sin ky)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (fma (* ky ky) -0.16666666666666666 1.0))
        (t_4 (hypot (sin ky) (sin kx))))
   (if (<= t_2 -1.0)
     t_1
     (if (<= t_2 -0.05)
       (/ (* (sin ky) th) t_4)
       (if (<= t_2 2e-5)
         (/ (* (* t_3 (sin th)) ky) (hypot (* t_3 ky) (sin kx)))
         (if (<= t_2 0.98)
           (* (/ (* (fma (* th th) -0.16666666666666666 1.0) th) t_4) (sin ky))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = fma((ky * ky), -0.16666666666666666, 1.0);
	double t_4 = hypot(sin(ky), sin(kx));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.05) {
		tmp = (sin(ky) * th) / t_4;
	} else if (t_2 <= 2e-5) {
		tmp = ((t_3 * sin(th)) * ky) / hypot((t_3 * ky), sin(kx));
	} else if (t_2 <= 0.98) {
		tmp = ((fma((th * th), -0.16666666666666666, 1.0) * th) / t_4) * sin(ky);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = fma(Float64(ky * ky), -0.16666666666666666, 1.0)
	t_4 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) * th) / t_4);
	elseif (t_2 <= 2e-5)
		tmp = Float64(Float64(Float64(t_3 * sin(th)) * ky) / hypot(Float64(t_3 * ky), sin(kx)));
	elseif (t_2 <= 0.98)
		tmp = Float64(Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / t_4) * sin(ky));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[th], $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[ky], $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[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], t$95$1, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$2, 2e-5], N[(N[(N[(t$95$3 * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(t$95$3 * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.98], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / t$95$4), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1 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 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000016e-5

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 93.3

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 93.3

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

   10. Applied rewrites 93.3%

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

   if 2.00000000000000016e-5 < (/.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 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

Alternative 9: 64.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.05)
     (* (/ 1.0 (sin ky)) (* (sin th) (fabs (sin ky))))
     (if (<= t_1 4e-193)
       (*
        (sqrt (/ 1.0 (- 1.0 (cos (* -2.0 kx)))))
        (* (* (sin th) ky) (sqrt 2.0)))
       (if (<= t_1 0.1) (* (/ (sin th) (sin kx)) (sin ky)) (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 <= -0.05) {
		tmp = (1.0 / sin(ky)) * (sin(th) * fabs(sin(ky)));
	} else if (t_1 <= 4e-193) {
		tmp = sqrt((1.0 / (1.0 - cos((-2.0 * kx))))) * ((sin(th) * ky) * sqrt(2.0));
	} else if (t_1 <= 0.1) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} 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) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.05d0)) then
        tmp = (1.0d0 / sin(ky)) * (sin(th) * abs(sin(ky)))
    else if (t_1 <= 4d-193) then
        tmp = sqrt((1.0d0 / (1.0d0 - cos(((-2.0d0) * kx))))) * ((sin(th) * ky) * sqrt(2.0d0))
    else if (t_1 <= 0.1d0) then
        tmp = (sin(th) / sin(kx)) * sin(ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

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 <= -0.05) {
		tmp = (1.0 / Math.sin(ky)) * (Math.sin(th) * Math.abs(Math.sin(ky)));
	} else if (t_1 <= 4e-193) {
		tmp = Math.sqrt((1.0 / (1.0 - Math.cos((-2.0 * kx))))) * ((Math.sin(th) * ky) * Math.sqrt(2.0));
	} else if (t_1 <= 0.1) {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	} else {
		tmp = 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 <= -0.05:
		tmp = (1.0 / math.sin(ky)) * (math.sin(th) * math.fabs(math.sin(ky)))
	elif t_1 <= 4e-193:
		tmp = math.sqrt((1.0 / (1.0 - math.cos((-2.0 * kx))))) * ((math.sin(th) * ky) * math.sqrt(2.0))
	elif t_1 <= 0.1:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	else:
		tmp = 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 <= -0.05)
		tmp = Float64(Float64(1.0 / sin(ky)) * Float64(sin(th) * abs(sin(ky))));
	elseif (t_1 <= 4e-193)
		tmp = Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * Float64(Float64(sin(th) * ky) * sqrt(2.0)));
	elseif (t_1 <= 0.1)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	else
		tmp = 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 <= -0.05)
		tmp = (1.0 / sin(ky)) * (sin(th) * abs(sin(ky)));
	elseif (t_1 <= 4e-193)
		tmp = sqrt((1.0 / (1.0 - cos((-2.0 * kx))))) * ((sin(th) * ky) * sqrt(2.0));
	elseif (t_1 <= 0.1)
		tmp = (sin(th) / sin(kx)) * sin(ky);
	else
		tmp = 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, -0.05], N[(N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-193], N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[Sin[th], $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))))) < -0.050000000000000003

   1. Initial program 97.3%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 3.8%

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

   10. Applied rewrites 56.6%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.0000000000000002e-193

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin th \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

   6. Applied rewrites 76.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      6. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(\color{blue}{-2} \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{2}\right) \]

      15. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\color{blue}{\sin th} \cdot ky\right) \cdot \sqrt{2}\right) \]

      16. lower-sqrt.f64 74.7

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \color{blue}{\sqrt{2}}\right) \]

   9. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)} \]

   if 4.0000000000000002e-193 < (/.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))))) < 0.10000000000000001

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.7

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.7

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th}}{\sin kx} \cdot \sin ky \]

      3. lower-sin.f64 57.3

         \[\leadsto \frac{\sin th}{\color{blue}{\sin kx}} \cdot \sin ky \]

   7. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]

   if 0.10000000000000001 < (/.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 89.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.1)
     (*
      (/ 1.0 (sin ky))
      (* (* (fma (* th th) -0.16666666666666666 1.0) th) (fabs (sin ky))))
     (if (<= t_1 4e-193)
       (*
        (sqrt (/ 1.0 (- 1.0 (cos (* -2.0 kx)))))
        (* (* (sin th) ky) (sqrt 2.0)))
       (if (<= t_1 0.1) (* (/ (sin th) (sin kx)) (sin ky)) (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 <= -0.1) {
		tmp = (1.0 / sin(ky)) * ((fma((th * th), -0.16666666666666666, 1.0) * th) * fabs(sin(ky)));
	} else if (t_1 <= 4e-193) {
		tmp = sqrt((1.0 / (1.0 - cos((-2.0 * kx))))) * ((sin(th) * ky) * sqrt(2.0));
	} else if (t_1 <= 0.1) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} 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 <= -0.1)
		tmp = Float64(Float64(1.0 / sin(ky)) * Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * abs(sin(ky))));
	elseif (t_1 <= 4e-193)
		tmp = Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * Float64(Float64(sin(th) * ky) * sqrt(2.0)));
	elseif (t_1 <= 0.1)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	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, -0.1], N[(N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-193], N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[Sin[th], $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))))) < -0.10000000000000001

   1. Initial program 97.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]

      14. lower-sin.f64 96.8

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin th \cdot \sin ky\right)} \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 3.8%

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]

   9. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot \sin \color{blue}{ky}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 4.1%

       \[\leadsto \frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \sin \color{blue}{ky}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 29.7%

       \[\leadsto \frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \left|\sin ky\right|\right) \]

   if -0.10000000000000001 < (/.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))))) < 4.0000000000000002e-193

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin th \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

   6. Applied rewrites 76.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      6. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(\color{blue}{-2} \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{2}\right) \]

      15. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\color{blue}{\sin th} \cdot ky\right) \cdot \sqrt{2}\right) \]

      16. lower-sqrt.f64 72.3

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \color{blue}{\sqrt{2}}\right) \]

   9. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)} \]

   if 4.0000000000000002e-193 < (/.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))))) < 0.10000000000000001

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.7

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.7

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th}}{\sin kx} \cdot \sin ky \]

      3. lower-sin.f64 57.3

         \[\leadsto \frac{\sin th}{\color{blue}{\sin kx}} \cdot \sin ky \]

   7. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]

   if 0.10000000000000001 < (/.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 89.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.1)
     (*
      (/ 1.0 (sin ky))
      (* (* (fma (* th th) -0.16666666666666666 1.0) th) (fabs (sin ky))))
     (if (<= t_1 4e-193)
       (*
        (sqrt (/ 1.0 (- 1.0 (cos (* -2.0 kx)))))
        (* (* (sin th) ky) (sqrt 2.0)))
       (if (<= t_1 0.1) (* (/ ky (sin kx)) (sin 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 <= -0.1) {
		tmp = (1.0 / sin(ky)) * ((fma((th * th), -0.16666666666666666, 1.0) * th) * fabs(sin(ky)));
	} else if (t_1 <= 4e-193) {
		tmp = sqrt((1.0 / (1.0 - cos((-2.0 * kx))))) * ((sin(th) * ky) * sqrt(2.0));
	} else if (t_1 <= 0.1) {
		tmp = (ky / sin(kx)) * sin(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 <= -0.1)
		tmp = Float64(Float64(1.0 / sin(ky)) * Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * abs(sin(ky))));
	elseif (t_1 <= 4e-193)
		tmp = Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * Float64(Float64(sin(th) * ky) * sqrt(2.0)));
	elseif (t_1 <= 0.1)
		tmp = Float64(Float64(ky / sin(kx)) * sin(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, -0.1], N[(N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-193], N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $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))))) < -0.10000000000000001

   1. Initial program 97.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]

      14. lower-sin.f64 96.8

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin th \cdot \sin ky\right)} \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 3.8%

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]

   9. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot \sin \color{blue}{ky}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 4.1%

       \[\leadsto \frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \sin \color{blue}{ky}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 29.7%

       \[\leadsto \frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \left|\sin ky\right|\right) \]

   if -0.10000000000000001 < (/.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))))) < 4.0000000000000002e-193

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin th \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

   6. Applied rewrites 76.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      6. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(\color{blue}{-2} \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{2}\right) \]

      15. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\color{blue}{\sin th} \cdot ky\right) \cdot \sqrt{2}\right) \]

      16. lower-sqrt.f64 72.3

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \color{blue}{\sqrt{2}}\right) \]

   9. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)} \]

   if 4.0000000000000002e-193 < (/.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))))) < 0.10000000000000001

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 57.3

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.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 89.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.1)
     (*
      (/ 1.0 (sin ky))
      (* (* (fma (* th th) -0.16666666666666666 1.0) th) (fabs (sin ky))))
     (if (<= t_1 0.1) (* (/ ky (sin kx)) (sin 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 <= -0.1) {
		tmp = (1.0 / sin(ky)) * ((fma((th * th), -0.16666666666666666, 1.0) * th) * fabs(sin(ky)));
	} else if (t_1 <= 0.1) {
		tmp = (ky / sin(kx)) * sin(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 <= -0.1)
		tmp = Float64(Float64(1.0 / sin(ky)) * Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * abs(sin(ky))));
	elseif (t_1 <= 0.1)
		tmp = Float64(Float64(ky / sin(kx)) * sin(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, -0.1], N[(N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $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))))) < -0.10000000000000001

   1. Initial program 97.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]

      14. lower-sin.f64 96.8

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin th \cdot \sin ky\right)} \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 3.8%

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]

   9. Taylor expanded in th around 0

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right) \cdot \sin \color{blue}{ky}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 4.1%

       \[\leadsto \frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \sin \color{blue}{ky}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 29.7%

       \[\leadsto \frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \left|\sin ky\right|\right) \]

   if -0.10000000000000001 < (/.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))))) < 0.10000000000000001

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 57.7

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.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 89.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.1)
   (* (/ ky (sin kx)) (sin 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)))) <= 0.1) {
		tmp = (ky / sin(kx)) * sin(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)))) <= 0.1d0) then
        tmp = (ky / sin(kx)) * sin(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)))) <= 0.1) {
		tmp = (ky / Math.sin(kx)) * Math.sin(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)))) <= 0.1:
		tmp = (ky / math.sin(kx)) * math.sin(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)))) <= 0.1)
		tmp = Float64(Float64(ky / sin(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) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.1)
		tmp = (ky / sin(kx)) * sin(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], 0.1], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[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))))) < 0.10000000000000001

   1. Initial program 98.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 32.9

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 32.9%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.10000000000000001 < (/.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 89.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.1)
   (/ (* (sin th) ky) (sin kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.1) {
		tmp = (sin(th) * ky) / sin(kx);
	} 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)))) <= 0.1d0) then
        tmp = (sin(th) * ky) / sin(kx)
    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)))) <= 0.1) {
		tmp = (Math.sin(th) * ky) / Math.sin(kx);
	} 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)))) <= 0.1:
		tmp = (math.sin(th) * ky) / math.sin(kx)
	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)))) <= 0.1)
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	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)))) <= 0.1)
		tmp = (sin(th) * ky) / sin(kx);
	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], 0.1], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $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))))) < 0.10000000000000001

   1. Initial program 98.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 98.5

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sin kx} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sin kx} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th} \cdot ky}{\sin kx} \]

      5. lower-sin.f64 31.4

         \[\leadsto \frac{\sin th \cdot ky}{\color{blue}{\sin kx}} \]

   7. Applied rewrites 31.4%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   if 0.10000000000000001 < (/.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 89.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.1:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-167}:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (* (sqrt (/ 1.0 (- 1.0 (cos (* -2.0 kx))))) (* (* (sin th) ky) (sqrt 2.0)))
   (if (<= (sin kx) 2e-167)
     (* (/ 1.0 (sin ky)) (* (sin th) (fabs (sin ky))))
     (if (<= (sin kx) 2e-6)
       (*
        (/ (sin ky) (sqrt (+ (* kx kx) (fma -0.5 (cos (* -2.0 ky)) 0.5))))
        (sin th))
       (* (/ (sin ky) (sin kx)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = sqrt((1.0 / (1.0 - cos((-2.0 * kx))))) * ((sin(th) * ky) * sqrt(2.0));
	} else if (sin(kx) <= 2e-167) {
		tmp = (1.0 / sin(ky)) * (sin(th) * fabs(sin(ky)));
	} else if (sin(kx) <= 2e-6) {
		tmp = (sin(ky) / sqrt(((kx * kx) + fma(-0.5, cos((-2.0 * ky)), 0.5)))) * sin(th);
	} else {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(sqrt(Float64(1.0 / Float64(1.0 - cos(Float64(-2.0 * kx))))) * Float64(Float64(sin(th) * ky) * sqrt(2.0)));
	elseif (sin(kx) <= 2e-167)
		tmp = Float64(Float64(1.0 / sin(ky)) * Float64(sin(th) * abs(sin(ky))));
	elseif (sin(kx) <= 2e-6)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + fma(-0.5, cos(Float64(-2.0 * ky)), 0.5)))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[Sqrt[N[(1.0 / N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-167], N[(N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-6], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(-0.5 * N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.0050000000000000001

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin th \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

   6. Applied rewrites 99.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{1 - \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      6. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(\color{blue}{-2} \cdot kx\right)}} \cdot \left(ky \cdot \left(\sin th \cdot \sqrt{2}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \color{blue}{\left(\left(ky \cdot \sin th\right) \cdot \sqrt{2}\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{2}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\color{blue}{\left(\sin th \cdot ky\right)} \cdot \sqrt{2}\right) \]

      15. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\color{blue}{\sin th} \cdot ky\right) \cdot \sqrt{2}\right) \]

      16. lower-sqrt.f64 50.1

          \[\leadsto \sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \color{blue}{\sqrt{2}}\right) \]

   9. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)} \]

   if -0.0050000000000000001 < (sin.f64 kx) < 2e-167

   1. Initial program 86.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]

      14. lower-sin.f64 83.5

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   5. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin th \cdot \sin ky\right)} \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 81.1%

       \[\leadsto \frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right) \]

   if 2e-167 < (sin.f64 kx) < 1.99999999999999991e-6

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 99.7

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      9. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 96.2

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   7. Applied rewrites 96.2%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   8. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}\right)}}} \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\color{blue}{\frac{-1}{2}} \cdot \cos \left(2 \cdot ky\right) + \frac{1}{2}\right)}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right), \frac{1}{2}\right)}}} \cdot \sin th \]

      5. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(\frac{-1}{2}, \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)}, \frac{1}{2}\right)}} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\color{blue}{-2} \cdot ky\right), \frac{1}{2}\right)}} \cdot \sin th \]

      9. lower-*.f64 96.2

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(-0.5, \cos \color{blue}{\left(-2 \cdot ky\right)}, 0.5\right)}} \cdot \sin th \]

   10. Applied rewrites 96.2%

       \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}}} \cdot \sin th \]

   if 1.99999999999999991e-6 < (sin.f64 kx)

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 54.2

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 54.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\sqrt{\frac{1}{1 - \cos \left(-2 \cdot kx\right)}} \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-167}:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(-0.5, \cos \left(-2 \cdot ky\right), 0.5\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-92}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-92)
   (* (pow th 3.0) -0.16666666666666666)
   (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)))) <= 5e-92) {
		tmp = pow(th, 3.0) * -0.16666666666666666;
	} 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)))) <= 5d-92) then
        tmp = (th ** 3.0d0) * (-0.16666666666666666d0)
    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)))) <= 5e-92) {
		tmp = Math.pow(th, 3.0) * -0.16666666666666666;
	} 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)))) <= 5e-92:
		tmp = math.pow(th, 3.0) * -0.16666666666666666
	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)))) <= 5e-92)
		tmp = Float64((th ^ 3.0) * -0.16666666666666666);
	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)))) <= 5e-92)
		tmp = (th ^ 3.0) * -0.16666666666666666;
	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], 5e-92], N[(N[Power[th, 3.0], $MachinePrecision] * -0.16666666666666666), $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))))) < 5.00000000000000011e-92

   1. Initial program 98.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.4%

      \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 13.9%

       \[\leadsto {th}^{3} \cdot -0.16666666666666666 \]

   if 5.00000000000000011e-92 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 53.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 53.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-92}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.35e-6)
   (*
    (/
     (sin th)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin ky))
   (*
    (/
     (sin ky)
     (/
      (sqrt
       (fma (- 1.0 (cos (* 2.0 ky))) 2.0 (* 2.0 (- 1.0 (cos (* 2.0 kx))))))
      2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.35e-6) {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	} else {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, (2.0 * (1.0 - cos((2.0 * kx)))))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.35e-6)
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 2.0, Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * kx)))))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.35e-6], N[(N[(N[Sin[th], $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[ky], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.34999999999999999e-6

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 93.8

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.7

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin ky \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin ky \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin ky \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin ky \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin ky \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin ky \]

      6. lower-*.f64 66.9

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin ky \]

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin ky \]

   if 1.34999999999999999e-6 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      8. frac-add N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      11. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{\color{blue}{4}}}} \cdot \sin th \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\color{blue}{2}}} \cdot \sin th \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{2}}} \cdot \sin th \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sin ky}{\sqrt{\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)}} \cdot \sqrt{2}\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.35e-6)
   (*
    (/
     (sin th)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin ky))
   (*
    (*
     (/
      (sin ky)
      (sqrt (+ (- 1.0 (cos (* -2.0 kx))) (- 1.0 (cos (* -2.0 ky))))))
     (sqrt 2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.35e-6) {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	} else {
		tmp = ((sin(ky) / sqrt(((1.0 - cos((-2.0 * kx))) + (1.0 - cos((-2.0 * ky)))))) * sqrt(2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.35e-6)
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky));
	else
		tmp = Float64(Float64(Float64(sin(ky) / sqrt(Float64(Float64(1.0 - cos(Float64(-2.0 * kx))) + Float64(1.0 - cos(Float64(-2.0 * ky)))))) * sqrt(2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.35e-6], N[(N[(N[Sin[th], $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[ky], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.34999999999999999e-6

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 93.8

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.7

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin ky \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin ky \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin ky \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin ky \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin ky \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin ky \]

      6. lower-*.f64 66.9

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin ky \]

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin ky \]

   if 1.34999999999999999e-6 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      8. div-add-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2}}}} \cdot \sin th \]

      9. sqrt-div N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

   6. Applied rewrites 98.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}}{\sqrt{2}}}} \cdot \sin th \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\left(\frac{\sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}\right)} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}\right)} \cdot \sin th \]

   8. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\left(\frac{\sin ky}{\sqrt{\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)}} \cdot \sqrt{2}\right)} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\left(\sin th \cdot \frac{\sin ky}{\sqrt{\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)}}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.35e-6)
   (*
    (/
     (sin th)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin ky))
   (*
    (*
     (sin th)
     (/
      (sin ky)
      (sqrt (+ (- 1.0 (cos (* -2.0 kx))) (- 1.0 (cos (* -2.0 ky)))))))
    (sqrt 2.0))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.35e-6) {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	} else {
		tmp = (sin(th) * (sin(ky) / sqrt(((1.0 - cos((-2.0 * kx))) + (1.0 - cos((-2.0 * ky))))))) * sqrt(2.0);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.35e-6)
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky));
	else
		tmp = Float64(Float64(sin(th) * Float64(sin(ky) / sqrt(Float64(Float64(1.0 - cos(Float64(-2.0 * kx))) + Float64(1.0 - cos(Float64(-2.0 * ky))))))) * sqrt(2.0));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.35e-6], N[(N[(N[Sin[th], $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[ky], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.34999999999999999e-6

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 93.8

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.7

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin ky \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin ky \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin ky \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin ky \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin ky \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin ky \]

      6. lower-*.f64 66.9

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin ky \]

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin ky \]

   if 1.34999999999999999e-6 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.4

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.4

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\sin th \cdot \frac{\sin ky}{\sqrt{\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)}}\right) \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 63.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right)\\ \mathbf{if}\;ky \leq 100:\\ \;\;\;\;\frac{\left(t\_1 \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(t\_1 \cdot ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fma (* ky ky) -0.16666666666666666 1.0)))
   (if (<= ky 100.0)
     (/ (* (* t_1 (sin th)) ky) (hypot (* t_1 ky) (sin kx)))
     (* (/ 1.0 (sin ky)) (* (sin th) (fabs (sin ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0);
	double tmp;
	if (ky <= 100.0) {
		tmp = ((t_1 * sin(th)) * ky) / hypot((t_1 * ky), sin(kx));
	} else {
		tmp = (1.0 / sin(ky)) * (sin(th) * fabs(sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(Float64(ky * ky), -0.16666666666666666, 1.0)
	tmp = 0.0
	if (ky <= 100.0)
		tmp = Float64(Float64(Float64(t_1 * sin(th)) * ky) / hypot(Float64(t_1 * ky), sin(kx)));
	else
		tmp = Float64(Float64(1.0 / sin(ky)) * Float64(sin(th) * abs(sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]}, If[LessEqual[ky, 100.0], N[(N[(N[(t$95$1 * N[Sin[th], $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(t$95$1 * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 100

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lower-*.f64 91.4

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \]

      14. lower-hypot.f64 94.5

          \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   4. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(\sin th + \frac{-1}{6} \cdot \left({ky}^{2} \cdot \sin th\right)\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sin th + \frac{-1}{6} \cdot \left({ky}^{2} \cdot \sin th\right)\right) \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\sin th + \frac{-1}{6} \cdot \color{blue}{\left(\sin th \cdot {ky}^{2}\right)}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\sin th + \color{blue}{\left(\frac{-1}{6} \cdot \sin th\right) \cdot {ky}^{2}}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sin th + \left(\frac{-1}{6} \cdot \sin th\right) \cdot {ky}^{2}\right) \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\sin th + \color{blue}{\frac{-1}{6} \cdot \left(\sin th \cdot {ky}^{2}\right)}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\sin th + \frac{-1}{6} \cdot \color{blue}{\left({ky}^{2} \cdot \sin th\right)}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\sin th + \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2}\right) \cdot \sin th}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      8. distribute-rgt1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot \sin th\right)} \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \sin th\right)} \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\left(\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      16. lower-sin.f64 55.2

          \[\leadsto \frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \color{blue}{\sin th}\right) \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   7. Applied rewrites 55.2%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \sin th\right) \cdot ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}, \sin kx\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}, \sin kx\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky, \sin kx\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\left(\color{blue}{{ky}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot ky, \sin kx\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right)} \cdot ky, \sin kx\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \]

      7. lower-*.f64 59.6

         \[\leadsto \frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \]

   10. Applied rewrites 59.6%

       \[\leadsto \frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}, \sin kx\right)} \]

   if 100 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}}} \cdot \left(\sin ky \cdot \sin th\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\sin kx}, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \color{blue}{\sin kx}, {\sin ky}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, \color{blue}{{\sin ky}^{2}}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\color{blue}{\sin ky}}^{2}\right)}} \cdot \left(\sin ky \cdot \sin th\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)} \]

      13. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]

      14. lower-sin.f64 99.4

          \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(\sin kx, \sin kx, {\sin ky}^{2}\right)}} \cdot \left(\sin th \cdot \sin ky\right)} \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 31.7%

      \[\leadsto \frac{1}{\sin ky} \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 58.8%

       \[\leadsto \frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 100:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \sin th\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin ky} \cdot \left(\sin th \cdot \left|\sin ky\right|\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 23.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (sin th))

C

double code(double kx, double ky, double th) {
	return 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(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th);
}

Python

def code(kx, ky, th):
	return math.sin(th)

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 95.5%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 23.0

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 23.0%

   \[\leadsto \color{blue}{\sin th} \]

6. Final simplification 23.0%

   \[\leadsto \sin th \]
7. Add Preprocessing

Alternative 22: 13.4% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (fma (* -0.16666666666666666 (* th th)) th th))

C

double code(double kx, double ky, double th) {
	return fma((-0.16666666666666666 * (th * th)), th, th);
}

Julia

function code(kx, ky, th)
	return fma(Float64(-0.16666666666666666 * Float64(th * th)), th, th)
end

Wolfram

code[kx_, ky_, th_] := N[(N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] * th + th), $MachinePrecision]

Derivation

1. Initial program 95.5%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 23.0

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 23.0%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 13.0%

   \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]
9. Step-by-step derivation

10. Applied rewrites 13.0%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \]

11. Final simplification 13.0%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025006 
(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)))