Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 8.2s

Alternatives: 26

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.7

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

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: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.105:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(th \cdot th\right) - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_3 \leq 0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, t\_1\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.999756666791005:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_1 t_2)))))
   (if (<= t_3 -0.996)
     (* (/ (sin ky) (sqrt t_2)) (sin th))
     (if (<= t_3 -0.105)
       (*
        (/ (sin ky) (hypot (sin ky) (sin kx)))
        (*
         (fma
          (-
           (*
            (fma (* th th) -0.0001984126984126984 0.008333333333333333)
            (* th th))
           0.16666666666666666)
          (* th th)
          1.0)
         th))
       (if (<= t_3 0.1)
         (*
          (/
           (sin ky)
           (sqrt
            (fma
             (fma
              (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
              (* ky ky)
              1.0)
             (* ky ky)
             t_1)))
          (sin th))
         (if (<= t_3 0.999756666791005)
           (* (* th (sin ky)) (/ 1.0 (hypot (sin kx) (sin ky))))
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((t_1 + t_2));
	double tmp;
	if (t_3 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_3 <= -0.105) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * (fma(((fma((th * th), -0.0001984126984126984, 0.008333333333333333) * (th * th)) - 0.16666666666666666), (th * th), 1.0) * th);
	} else if (t_3 <= 0.1) {
		tmp = (sin(ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), t_1))) * sin(th);
	} else if (t_3 <= 0.999756666791005) {
		tmp = (th * sin(ky)) * (1.0 / hypot(sin(kx), sin(ky)));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_3 <= -0.105)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * Float64(fma(Float64(Float64(fma(Float64(th * th), -0.0001984126984126984, 0.008333333333333333) * Float64(th * th)) - 0.16666666666666666), Float64(th * th), 1.0) * th));
	elseif (t_3 <= 0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), t_1))) * sin(th));
	elseif (t_3 <= 0.999756666791005)
		tmp = Float64(Float64(th * sin(ky)) * Float64(1.0 / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $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[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.105], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(th * th), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.999756666791005], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 85.1%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.104999999999999996

   1. Initial program 99.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 49.7%

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

   if -0.104999999999999996 < (/.f64 (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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-pow.f64 96.2

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

   5. Applied rewrites 96.2%

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

   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))))) < 0.999756666791005055

   1. Initial program 99.4%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow-to-exp N/A

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

      6. sinh-+-cosh-rev N/A

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

      7. lower-+.f64 N/A

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. lift-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 48.9

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

   10. Applied rewrites 48.9%

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

   if 0.999756666791005055 < (/.f64 (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 79.6%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 100.0%

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

Alternative 3: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.105:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(th \cdot th\right) - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.008333333333333333 - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.999756666791005:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.996)
     (* (/ (sin ky) (sqrt t_1)) (sin th))
     (if (<= t_2 -0.105)
       (*
        (/ (sin ky) (hypot (sin ky) (sin kx)))
        (*
         (fma
          (-
           (*
            (fma (* th th) -0.0001984126984126984 0.008333333333333333)
            (* th th))
           0.16666666666666666)
          (* th th)
          1.0)
         th))
       (if (<= t_2 0.1)
         (*
          (/
           (sin ky)
           (hypot
            (*
             (fma
              (- (* (* ky ky) 0.008333333333333333) 0.16666666666666666)
              (* ky ky)
              1.0)
             ky)
            (sin kx)))
          (sin th))
         (if (<= t_2 0.999756666791005)
           (* (* th (sin ky)) (/ 1.0 (hypot (sin kx) (sin ky))))
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	} else if (t_2 <= -0.105) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * (fma(((fma((th * th), -0.0001984126984126984, 0.008333333333333333) * (th * th)) - 0.16666666666666666), (th * th), 1.0) * th);
	} else if (t_2 <= 0.1) {
		tmp = (sin(ky) / hypot((fma((((ky * ky) * 0.008333333333333333) - 0.16666666666666666), (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else if (t_2 <= 0.999756666791005) {
		tmp = (th * sin(ky)) * (1.0 / hypot(sin(kx), sin(ky)));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * sin(th));
	elseif (t_2 <= -0.105)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * Float64(fma(Float64(Float64(fma(Float64(th * th), -0.0001984126984126984, 0.008333333333333333) * Float64(th * th)) - 0.16666666666666666), Float64(th * th), 1.0) * th));
	elseif (t_2 <= 0.1)
		tmp = Float64(Float64(sin(ky) / hypot(Float64(fma(Float64(Float64(Float64(ky * ky) * 0.008333333333333333) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	elseif (t_2 <= 0.999756666791005)
		tmp = Float64(Float64(th * sin(ky)) * Float64(1.0 / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.105], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(th * th), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(th * th), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.999756666791005], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 85.1%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.104999999999999996

   1. Initial program 99.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 49.7%

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

   if -0.104999999999999996 < (/.f64 (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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

   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))))) < 0.999756666791005055

   1. Initial program 99.4%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow-to-exp N/A

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

      6. sinh-+-cosh-rev N/A

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

      7. lower-+.f64 N/A

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. lift-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 48.9

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

   10. Applied rewrites 48.9%

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

   if 0.999756666791005055 < (/.f64 (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 79.6%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 100.0%

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

Alternative 4: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.105:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.008333333333333333 - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.999756666791005:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.996)
     (* (/ (sin ky) (sqrt t_1)) (sin th))
     (if (<= t_2 -0.105)
       (*
        (/ (sin ky) (hypot (sin ky) (sin kx)))
        (*
         (fma
          (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
          (* th th)
          1.0)
         th))
       (if (<= t_2 0.1)
         (*
          (/
           (sin ky)
           (hypot
            (*
             (fma
              (- (* (* ky ky) 0.008333333333333333) 0.16666666666666666)
              (* ky ky)
              1.0)
             ky)
            (sin kx)))
          (sin th))
         (if (<= t_2 0.999756666791005)
           (* (* th (sin ky)) (/ 1.0 (hypot (sin kx) (sin ky))))
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	} else if (t_2 <= -0.105) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * (fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th);
	} else if (t_2 <= 0.1) {
		tmp = (sin(ky) / hypot((fma((((ky * ky) * 0.008333333333333333) - 0.16666666666666666), (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else if (t_2 <= 0.999756666791005) {
		tmp = (th * sin(ky)) * (1.0 / hypot(sin(kx), sin(ky)));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * sin(th));
	elseif (t_2 <= -0.105)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th));
	elseif (t_2 <= 0.1)
		tmp = Float64(Float64(sin(ky) / hypot(Float64(fma(Float64(Float64(Float64(ky * ky) * 0.008333333333333333) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	elseif (t_2 <= 0.999756666791005)
		tmp = Float64(Float64(th * sin(ky)) * Float64(1.0 / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.105], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.999756666791005], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 85.1%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.104999999999999996

   1. Initial program 99.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 50.0

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

   7. Applied rewrites 50.0%

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

   if -0.104999999999999996 < (/.f64 (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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

   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))))) < 0.999756666791005055

   1. Initial program 99.4%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow-to-exp N/A

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

      6. sinh-+-cosh-rev N/A

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

      7. lower-+.f64 N/A

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. lift-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 48.9

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

   10. Applied rewrites 48.9%

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

   if 0.999756666791005055 < (/.f64 (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 79.6%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 100.0%

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

Alternative 5: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.008333333333333333 - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.999756666791005:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.996)
     (* (/ (sin ky) (sqrt t_1)) (sin th))
     (if (<= t_2 -0.1)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_2 0.1)
         (*
          (/
           (sin ky)
           (hypot
            (*
             (fma
              (- (* (* ky ky) 0.008333333333333333) 0.16666666666666666)
              (* ky ky)
              1.0)
             ky)
            (sin kx)))
          (sin th))
         (if (<= t_2 0.999756666791005)
           (* (* th (sin ky)) (/ 1.0 (hypot (sin kx) (sin ky))))
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_1)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_2 <= 0.1) {
		tmp = (sin(ky) / hypot((fma((((ky * ky) * 0.008333333333333333) - 0.16666666666666666), (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else if (t_2 <= 0.999756666791005) {
		tmp = (th * sin(ky)) * (1.0 / hypot(sin(kx), sin(ky)));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_2 <= 0.1)
		tmp = Float64(Float64(sin(ky) / hypot(Float64(fma(Float64(Float64(Float64(ky * ky) * 0.008333333333333333) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	elseif (t_2 <= 0.999756666791005)
		tmp = Float64(Float64(th * sin(ky)) * Float64(1.0 / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.999756666791005], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 85.1%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -0.996 < (/.f64 (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 99.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 49.0%

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

   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))))) < 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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 97.3

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

   7. Applied rewrites 97.3%

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

   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))))) < 0.999756666791005055

   1. Initial program 99.4%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow-to-exp N/A

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

      6. sinh-+-cosh-rev N/A

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

      7. lower-+.f64 N/A

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. lift-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 48.9

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

   10. Applied rewrites 48.9%

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

   if 0.999756666791005055 < (/.f64 (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 79.6%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 100.0%

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

Alternative 6: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.105:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_3 \leq 0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, t\_1\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.999756666791005:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_1 t_2)))))
   (if (<= t_3 -0.996)
     (* (/ (sin ky) (sqrt t_2)) (sin th))
     (if (<= t_3 -0.105)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_3 0.1)
         (* (/ (sin ky) (sqrt (fma ky ky t_1))) (sin th))
         (if (<= t_3 0.999756666791005)
           (* (* th (sin ky)) (/ 1.0 (hypot (sin kx) (sin ky))))
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((t_1 + t_2));
	double tmp;
	if (t_3 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_3 <= -0.105) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_3 <= 0.1) {
		tmp = (sin(ky) / sqrt(fma(ky, ky, t_1))) * sin(th);
	} else if (t_3 <= 0.999756666791005) {
		tmp = (th * sin(ky)) * (1.0 / hypot(sin(kx), sin(ky)));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_3 <= -0.105)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_3 <= 0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(ky, ky, t_1))) * sin(th));
	elseif (t_3 <= 0.999756666791005)
		tmp = Float64(Float64(th * sin(ky)) * Float64(1.0 / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $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[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.105], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * ky + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.999756666791005], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 85.1%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -0.996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.104999999999999996

   1. Initial program 99.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 50.0%

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

   if -0.104999999999999996 < (/.f64 (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

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 96.1

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

   5. Applied rewrites 96.1%

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

   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))))) < 0.999756666791005055

   1. Initial program 99.4%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow-to-exp N/A

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

      6. sinh-+-cosh-rev N/A

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

      7. lower-+.f64 N/A

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. lift-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 48.9

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

   10. Applied rewrites 48.9%

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

   if 0.999756666791005055 < (/.f64 (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 79.6%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 100.0%

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

Alternative 7: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.996:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{t\_1 + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.999756666791005:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_1 t_2)))))
   (if (<= t_3 -0.996)
     (* (/ (sin ky) (sqrt t_2)) (sin th))
     (if (<= t_3 -0.1)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_3 0.05)
         (*
          (/
           (*
            (fma
             (-
              (*
               (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
               (* ky ky))
              0.16666666666666666)
             (* ky ky)
             1.0)
            ky)
           (sqrt
            (+
             t_1
             (*
              (fma
               (-
                (*
                 (fma (* ky ky) -0.0031746031746031746 0.044444444444444446)
                 (* ky ky))
                0.3333333333333333)
               (* ky ky)
               1.0)
              (* ky ky)))))
          (sin th))
         (if (<= t_3 0.999756666791005)
           (* (* th (sin ky)) (/ 1.0 (hypot (sin kx) (sin ky))))
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((t_1 + t_2));
	double tmp;
	if (t_3 <= -0.996) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_3 <= 0.05) {
		tmp = ((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt((t_1 + (fma(((fma((ky * ky), -0.0031746031746031746, 0.044444444444444446) * (ky * ky)) - 0.3333333333333333), (ky * ky), 1.0) * (ky * ky))))) * sin(th);
	} else if (t_3 <= 0.999756666791005) {
		tmp = (th * sin(ky)) * (1.0 / hypot(sin(kx), sin(ky)));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.996)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(Float64(t_1 + Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0031746031746031746, 0.044444444444444446) * Float64(ky * ky)) - 0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))))) * sin(th));
	elseif (t_3 <= 0.999756666791005)
		tmp = Float64(Float64(th * sin(ky)) * Float64(1.0 / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $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[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.996], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[(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] / N[Sqrt[N[(t$95$1 + N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.999756666791005], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.996

   1. Initial program 85.1%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -0.996 < (/.f64 (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 99.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 49.0%

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

   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))))) < 0.050000000000000003

   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

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 97.1%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   8. Applied rewrites 97.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   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))))) < 0.999756666791005055

   1. Initial program 99.4%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow-to-exp N/A

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

      6. sinh-+-cosh-rev N/A

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

      7. lower-+.f64 N/A

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. lift-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 48.9

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

   10. Applied rewrites 48.9%

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

   if 0.999756666791005055 < (/.f64 (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 79.6%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 100.0%

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

Alternative 8: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.999:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{t\_2 + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.999756666791005:\\ \;\;\;\;\left(th \cdot \sin ky\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.999)
     t_1
     (if (<= t_3 -0.1)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_3 0.05)
         (*
          (/
           (*
            (fma
             (-
              (*
               (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
               (* ky ky))
              0.16666666666666666)
             (* ky ky)
             1.0)
            ky)
           (sqrt
            (+
             t_2
             (*
              (fma
               (-
                (*
                 (fma (* ky ky) -0.0031746031746031746 0.044444444444444446)
                 (* ky ky))
                0.3333333333333333)
               (* ky ky)
               1.0)
              (* ky ky)))))
          (sin th))
         (if (<= t_3 0.999756666791005)
           (* (* th (sin ky)) (/ 1.0 (hypot (sin kx) (sin ky))))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.999) {
		tmp = t_1;
	} else if (t_3 <= -0.1) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_3 <= 0.05) {
		tmp = ((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt((t_2 + (fma(((fma((ky * ky), -0.0031746031746031746, 0.044444444444444446) * (ky * ky)) - 0.3333333333333333), (ky * ky), 1.0) * (ky * ky))))) * sin(th);
	} else if (t_3 <= 0.999756666791005) {
		tmp = (th * sin(ky)) * (1.0 / hypot(sin(kx), sin(ky)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.999)
		tmp = t_1;
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(Float64(t_2 + Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0031746031746031746, 0.044444444444444446) * Float64(ky * ky)) - 0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))))) * sin(th));
	elseif (t_3 <= 0.999756666791005)
		tmp = Float64(Float64(th * sin(ky)) * Float64(1.0 / hypot(sin(kx), sin(ky))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.999], t$95$1, If[LessEqual[t$95$3, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[(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] / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.999756666791005], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $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))))) < -0.998999999999999999 or 0.999756666791005055 < (/.f64 (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 82.3%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 99.0%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 46.9%

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

   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))))) < 0.050000000000000003

   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

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 97.1%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   8. Applied rewrites 97.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   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))))) < 0.999756666791005055

   1. Initial program 99.4%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow-to-exp N/A

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

      6. sinh-+-cosh-rev N/A

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

      7. lower-+.f64 N/A

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. lift-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 48.9

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

   10. Applied rewrites 48.9%

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

Alternative 9: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.999:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_3 \leq 0.002:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{t\_2 + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.999756666791005:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.999)
     t_1
     (if (<= t_3 -0.1)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_3 0.002)
         (*
          (/
           (*
            (fma
             (-
              (*
               (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
               (* ky ky))
              0.16666666666666666)
             (* ky ky)
             1.0)
            ky)
           (sqrt
            (+
             t_2
             (*
              (fma
               (-
                (*
                 (fma (* ky ky) -0.0031746031746031746 0.044444444444444446)
                 (* ky ky))
                0.3333333333333333)
               (* ky ky)
               1.0)
              (* ky ky)))))
          (sin th))
         (if (<= t_3 0.999756666791005)
           (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.999) {
		tmp = t_1;
	} else if (t_3 <= -0.1) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_3 <= 0.002) {
		tmp = ((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt((t_2 + (fma(((fma((ky * ky), -0.0031746031746031746, 0.044444444444444446) * (ky * ky)) - 0.3333333333333333), (ky * ky), 1.0) * (ky * ky))))) * sin(th);
	} else if (t_3 <= 0.999756666791005) {
		tmp = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.999)
		tmp = t_1;
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_3 <= 0.002)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(Float64(t_2 + Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0031746031746031746, 0.044444444444444446) * Float64(ky * ky)) - 0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))))) * sin(th));
	elseif (t_3 <= 0.999756666791005)
		tmp = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.999], t$95$1, If[LessEqual[t$95$3, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 0.002], N[(N[(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] / N[Sqrt[N[(t$95$2 + N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.999756666791005], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $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))))) < -0.998999999999999999 or 0.999756666791005055 < (/.f64 (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 82.3%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 99.0%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 99.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 46.9%

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

   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))))) < 2e-3

   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

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 97.1%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   8. Applied rewrites 97.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   if 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))))) < 0.999756666791005055

   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 48.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. associate-*l/ N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

   9. Applied rewrites 48.8%

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

Alternative 10: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_4 \leq -0.999:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.002:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{t\_3 + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.999756666791005:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (* (/ (sin ky) (hypot (sin ky) (sin kx))) th))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin ky) 2.0))))))
   (if (<= t_4 -0.999)
     t_1
     (if (<= t_4 -0.1)
       t_2
       (if (<= t_4 0.002)
         (*
          (/
           (*
            (fma
             (-
              (*
               (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
               (* ky ky))
              0.16666666666666666)
             (* ky ky)
             1.0)
            ky)
           (sqrt
            (+
             t_3
             (*
              (fma
               (-
                (*
                 (fma (* ky ky) -0.0031746031746031746 0.044444444444444446)
                 (* ky ky))
                0.3333333333333333)
               (* ky ky)
               1.0)
              (* ky ky)))))
          (sin th))
         (if (<= t_4 0.999756666791005) t_2 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	double t_2 = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_4 <= -0.999) {
		tmp = t_1;
	} else if (t_4 <= -0.1) {
		tmp = t_2;
	} else if (t_4 <= 0.002) {
		tmp = ((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt((t_3 + (fma(((fma((ky * ky), -0.0031746031746031746, 0.044444444444444446) * (ky * ky)) - 0.3333333333333333), (ky * ky), 1.0) * (ky * ky))))) * sin(th);
	} else if (t_4 <= 0.999756666791005) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th))
	t_2 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th)
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.999)
		tmp = t_1;
	elseif (t_4 <= -0.1)
		tmp = t_2;
	elseif (t_4 <= 0.002)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(Float64(t_3 + Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0031746031746031746, 0.044444444444444446) * Float64(ky * ky)) - 0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))))) * sin(th));
	elseif (t_4 <= 0.999756666791005)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.999], t$95$1, If[LessEqual[t$95$4, -0.1], t$95$2, If[LessEqual[t$95$4, 0.002], N[(N[(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] / N[Sqrt[N[(t$95$3 + N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.999756666791005], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999 or 0.999756666791005055 < (/.f64 (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 82.3%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 99.0%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 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))))) < 0.999756666791005055

   1. Initial program 99.3%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 47.8%

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

   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))))) < 2e-3

   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

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 97.1%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   8. Applied rewrites 97.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 54.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot th\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -1.0)
     (* (/ (sin ky) (hypot (sin ky) kx)) th)
     (if (<= t_1 2e-58)
       (* (/ (sin ky) (sin kx)) (sin th))
       (if (<= t_1 0.002)
         (*
          (/
           ky
           (sqrt
            (+
             (- 0.5 (* 0.5 (cos (+ kx kx))))
             (*
              (fma
               (-
                (*
                 (fma (* ky ky) -0.0031746031746031746 0.044444444444444446)
                 (* ky ky))
                0.3333333333333333)
               (* ky ky)
               1.0)
              (* ky ky)))))
          (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 <= -1.0) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * th;
	} else if (t_1 <= 2e-58) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.002) {
		tmp = (ky / sqrt(((0.5 - (0.5 * cos((kx + kx)))) + (fma(((fma((ky * ky), -0.0031746031746031746, 0.044444444444444446) * (ky * ky)) - 0.3333333333333333), (ky * ky), 1.0) * (ky * ky))))) * 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 <= -1.0)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * th);
	elseif (t_1 <= 2e-58)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.002)
		tmp = Float64(Float64(ky / sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx)))) + Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0031746031746031746, 0.044444444444444446) * Float64(ky * ky)) - 0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))))) * 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, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 2e-58], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(N[(ky / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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))))) < -1

   1. Initial program 84.3%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 56.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.0000000000000001e-58

   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. lift-sin.f64 39.6

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

   5. Applied rewrites 39.6%

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

   if 2.0000000000000001e-58 < (/.f64 (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 99.4%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 96.4%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      7. cos-2 N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos kx \cdot \cos kx - \sin kx \cdot \sin kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      8. cos-sum N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      10. lower-+.f64 82.9

          \[\leadsto \frac{ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   10. Applied rewrites 82.9%

       \[\leadsto \frac{ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   if 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)))))

   1. Initial program 88.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 60.7

         \[\leadsto \sin th \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 65.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot th\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{t\_1 + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_1 t_2)))))
   (if (<= t_3 -0.1)
     (* (/ (sin ky) (sqrt t_2)) th)
     (if (<= t_3 0.05)
       (*
        (/
         (*
          (fma
           (-
            (*
             (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
             (* ky ky))
            0.16666666666666666)
           (* ky ky)
           1.0)
          ky)
         (sqrt
          (+
           t_1
           (*
            (fma
             (-
              (*
               (fma (* ky ky) -0.0031746031746031746 0.044444444444444446)
               (* ky ky))
              0.3333333333333333)
             (* ky ky)
             1.0)
            (* ky ky)))))
        (sin th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((t_1 + t_2));
	double tmp;
	if (t_3 <= -0.1) {
		tmp = (sin(ky) / sqrt(t_2)) * th;
	} else if (t_3 <= 0.05) {
		tmp = ((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt((t_1 + (fma(((fma((ky * ky), -0.0031746031746031746, 0.044444444444444446) * (ky * ky)) - 0.3333333333333333), (ky * ky), 1.0) * (ky * ky))))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * th);
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(Float64(t_1 + Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0031746031746031746, 0.044444444444444446) * Float64(ky * ky)) - 0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $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[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[(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] / N[Sqrt[N[(t$95$1 + N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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 91.2%

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

   3. Taylor expanded in ky around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 11.3

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

   5. Applied rewrites 11.3%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 6.9%

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

   9. Taylor expanded in kx around 0

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

       1. lower-pow.f64 N/A

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

       2. lift-sin.f64 30.1

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

   11. Applied rewrites 30.1%

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

   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))))) < 0.050000000000000003

   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

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 97.1%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   8. Applied rewrites 97.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

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

   1. Initial program 88.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 60.7

         \[\leadsto \sin th \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 65.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot th\\ \mathbf{elif}\;t\_3 \leq 0.088:\\ \;\;\;\;\frac{ky}{\sqrt{\mathsf{fma}\left(ky, ky, t\_1\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_1 t_2)))))
   (if (<= t_3 -0.1)
     (* (/ (sin ky) (sqrt t_2)) th)
     (if (<= t_3 0.088) (* (/ ky (sqrt (fma ky ky t_1))) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((t_1 + t_2));
	double tmp;
	if (t_3 <= -0.1) {
		tmp = (sin(ky) / sqrt(t_2)) * th;
	} else if (t_3 <= 0.088) {
		tmp = (ky / sqrt(fma(ky, ky, t_1))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * th);
	elseif (t_3 <= 0.088)
		tmp = Float64(Float64(ky / sqrt(fma(ky, ky, t_1))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $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[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 0.088], N[(N[(ky / N[Sqrt[N[(ky * ky + t$95$1), $MachinePrecision]], $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 91.2%

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

   3. Taylor expanded in ky around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 11.3

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

   5. Applied rewrites 11.3%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 6.9%

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

   9. Taylor expanded in kx around 0

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

       1. lower-pow.f64 N/A

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

       2. lift-sin.f64 30.1

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

   11. Applied rewrites 30.1%

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

   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))))) < 0.087999999999999995

   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

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 97.1%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 96.6%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   9. Taylor expanded in ky around 0

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

       1. pow2 N/A

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

       2. lower-fma.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-pow.f64 96.8

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

   11. Applied rewrites 96.8%

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

   if 0.087999999999999995 < (/.f64 (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 88.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 60.7

         \[\leadsto \sin th \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot th\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{ky}{\sqrt{t\_1}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_1 t_2)))))
   (if (<= t_3 -0.1)
     (* (/ (sin ky) (sqrt t_2)) th)
     (if (<= t_3 0.05) (* (/ ky (sqrt t_1)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((t_1 + t_2));
	double tmp;
	if (t_3 <= -0.1) {
		tmp = (sin(ky) / sqrt(t_2)) * th;
	} else if (t_3 <= 0.05) {
		tmp = (ky / sqrt(t_1)) * 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sin(kx) ** 2.0d0
    t_2 = sin(ky) ** 2.0d0
    t_3 = sin(ky) / sqrt((t_1 + t_2))
    if (t_3 <= (-0.1d0)) then
        tmp = (sin(ky) / sqrt(t_2)) * th
    else if (t_3 <= 0.05d0) then
        tmp = (ky / sqrt(t_1)) * sin(th)
    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.pow(Math.sin(kx), 2.0);
	double t_2 = Math.pow(Math.sin(ky), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((t_1 + t_2));
	double tmp;
	if (t_3 <= -0.1) {
		tmp = (Math.sin(ky) / Math.sqrt(t_2)) * th;
	} else if (t_3 <= 0.05) {
		tmp = (ky / Math.sqrt(t_1)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	t_2 = math.pow(math.sin(ky), 2.0)
	t_3 = math.sin(ky) / math.sqrt((t_1 + t_2))
	tmp = 0
	if t_3 <= -0.1:
		tmp = (math.sin(ky) / math.sqrt(t_2)) * th
	elif t_3 <= 0.05:
		tmp = (ky / math.sqrt(t_1)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * th);
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(ky / sqrt(t_1)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	t_2 = sin(ky) ^ 2.0;
	t_3 = sin(ky) / sqrt((t_1 + t_2));
	tmp = 0.0;
	if (t_3 <= -0.1)
		tmp = (sin(ky) / sqrt(t_2)) * th;
	elseif (t_3 <= 0.05)
		tmp = (ky / sqrt(t_1)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $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[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 0.05], N[(N[(ky / N[Sqrt[t$95$1], $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 91.2%

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

   3. Taylor expanded in ky around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 11.3

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

   5. Applied rewrites 11.3%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 6.9%

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

   9. Taylor expanded in kx around 0

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

       1. lower-pow.f64 N/A

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

       2. lift-sin.f64 30.1

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

   11. Applied rewrites 30.1%

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

   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))))) < 0.050000000000000003

   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

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 96.4

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in ky around 0

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

   8. Applied rewrites 95.8%

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

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

   1. Initial program 88.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 60.7

         \[\leadsto \sin th \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 65.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.1)
     (* (/ (sin ky) (hypot (sin ky) kx)) th)
     (if (<= t_2 0.05) (* (/ ky (sqrt t_1)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * th;
	} else if (t_2 <= 0.05) {
		tmp = (ky / sqrt(t_1)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((t_1 + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * th;
	} else if (t_2 <= 0.05) {
		tmp = (ky / Math.sqrt(t_1)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	t_2 = math.sin(ky) / math.sqrt((t_1 + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.1:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * th
	elif t_2 <= 0.05:
		tmp = (ky / math.sqrt(t_1)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * th);
	elseif (t_2 <= 0.05)
		tmp = Float64(Float64(ky / sqrt(t_1)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	t_2 = sin(ky) / sqrt((t_1 + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.1)
		tmp = (sin(ky) / hypot(sin(ky), kx)) * th;
	elseif (t_2 <= 0.05)
		tmp = (ky / sqrt(t_1)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.05], N[(N[(ky / N[Sqrt[t$95$1], $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 91.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.8%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot th \]
   9. Step-by-step derivation

   10. Applied rewrites 33.0%

       \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot th \]

   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))))) < 0.050000000000000003

   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

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      2. lift-pow.f64 96.4

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \cdot \sin th \]

   5. Applied rewrites 96.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 95.8%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 60.7

         \[\leadsto \sin th \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 46.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 2e-58)
     (* (/ (sin ky) (sin kx)) (sin th))
     (if (<= t_1 0.002)
       (*
        (/
         ky
         (sqrt
          (+
           (- 0.5 (* 0.5 (cos (+ kx kx))))
           (*
            (fma
             (-
              (*
               (fma (* ky ky) -0.0031746031746031746 0.044444444444444446)
               (* ky ky))
              0.3333333333333333)
             (* ky ky)
             1.0)
            (* ky ky)))))
        (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 <= 2e-58) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.002) {
		tmp = (ky / sqrt(((0.5 - (0.5 * cos((kx + kx)))) + (fma(((fma((ky * ky), -0.0031746031746031746, 0.044444444444444446) * (ky * ky)) - 0.3333333333333333), (ky * ky), 1.0) * (ky * ky))))) * 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 <= 2e-58)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.002)
		tmp = Float64(Float64(ky / sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx)))) + Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0031746031746031746, 0.044444444444444446) * Float64(ky * ky)) - 0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))))) * 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, 2e-58], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(N[(ky / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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))))) < 2.0000000000000001e-58

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-sin.f64 28.8

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \sin th \]

   5. Applied rewrites 28.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 2.0000000000000001e-58 < (/.f64 (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 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 96.4%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      7. cos-2 N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos kx \cdot \cos kx - \sin kx \cdot \sin kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      8. cos-sum N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      10. lower-+.f64 82.9

          \[\leadsto \frac{ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   10. Applied rewrites 82.9%

       \[\leadsto \frac{ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   if 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)))))

   1. Initial program 88.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 60.7

         \[\leadsto \sin th \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 45.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 2e-58)
     (* (/ ky (sin kx)) (sin th))
     (if (<= t_1 0.002)
       (*
        (/
         ky
         (sqrt
          (+
           (- 0.5 (* 0.5 (cos (+ kx kx))))
           (*
            (fma
             (-
              (*
               (fma (* ky ky) -0.0031746031746031746 0.044444444444444446)
               (* ky ky))
              0.3333333333333333)
             (* ky ky)
             1.0)
            (* ky ky)))))
        (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 <= 2e-58) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_1 <= 0.002) {
		tmp = (ky / sqrt(((0.5 - (0.5 * cos((kx + kx)))) + (fma(((fma((ky * ky), -0.0031746031746031746, 0.044444444444444446) * (ky * ky)) - 0.3333333333333333), (ky * ky), 1.0) * (ky * ky))))) * 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 <= 2e-58)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_1 <= 0.002)
		tmp = Float64(Float64(ky / sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx)))) + Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0031746031746031746, 0.044444444444444446) * Float64(ky * ky)) - 0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))))) * 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, 2e-58], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(N[(ky / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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))))) < 2.0000000000000001e-58

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 26.1

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   5. Applied rewrites 26.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 2.0000000000000001e-58 < (/.f64 (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 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 96.4%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      7. cos-2 N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos kx \cdot \cos kx - \sin kx \cdot \sin kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      8. cos-sum N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(kx + kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

      10. lower-+.f64 82.9

          \[\leadsto \frac{ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   10. Applied rewrites 82.9%

       \[\leadsto \frac{ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   if 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)))))

   1. Initial program 88.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 60.7

         \[\leadsto \sin th \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 44.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-11)
   (* (/ 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)))) <= 5e-11) {
		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)))) <= 5d-11) 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)))) <= 5e-11) {
		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)))) <= 5e-11:
		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)))) <= 5e-11)
		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)))) <= 5e-11)
		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], 5e-11], 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))))) < 5.00000000000000018e-11

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 27.1

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   5. Applied rewrites 27.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 5.00000000000000018e-11 < (/.f64 (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 88.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 59.5

         \[\leadsto \sin th \]

   5. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-11)
   (/ (* (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)))) <= 5e-11) {
		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)))) <= 5d-11) 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)))) <= 5e-11) {
		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)))) <= 5e-11:
		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)))) <= 5e-11)
		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)))) <= 5e-11)
		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], 5e-11], 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))))) < 5.00000000000000018e-11

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sin kx}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin \color{blue}{kx}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin \color{blue}{kx}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      5. lift-sin.f64 26.6

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   5. Applied rewrites 26.6%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   if 5.00000000000000018e-11 < (/.f64 (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 88.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 59.5

         \[\leadsto \sin th \]

   5. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 42.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{ky}{\sqrt{\mathsf{fma}\left(\left(kx \cdot kx\right) \cdot 0.044444444444444446 - 0.3333333333333333, kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.05)
   (*
    (/
     ky
     (sqrt
      (+
       (*
        (fma
         (- (* (* kx kx) 0.044444444444444446) 0.3333333333333333)
         (* kx kx)
         1.0)
        (* kx kx))
       (*
        (fma
         (-
          (*
           (fma (* ky ky) -0.0031746031746031746 0.044444444444444446)
           (* ky ky))
          0.3333333333333333)
         (* ky ky)
         1.0)
        (* ky ky)))))
    (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)))) <= 0.05) {
		tmp = (ky / sqrt(((fma((((kx * kx) * 0.044444444444444446) - 0.3333333333333333), (kx * kx), 1.0) * (kx * kx)) + (fma(((fma((ky * ky), -0.0031746031746031746, 0.044444444444444446) * (ky * ky)) - 0.3333333333333333), (ky * ky), 1.0) * (ky * ky))))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.05)
		tmp = Float64(Float64(ky / sqrt(Float64(Float64(fma(Float64(Float64(Float64(kx * kx) * 0.044444444444444446) - 0.3333333333333333), Float64(kx * kx), 1.0) * Float64(kx * kx)) + Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0031746031746031746, 0.044444444444444446) * Float64(ky * ky)) - 0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(ky / N[Sqrt[N[(N[(N[(N[(N[(N[(kx * kx), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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))))) < 0.050000000000000003

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 49.4%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{ky}{\sqrt{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot \color{blue}{{kx}^{2}} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot \color{blue}{{kx}^{2}} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       3. +-commutative N/A

          \[\leadsto \frac{ky}{\sqrt{\left({kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) + 1\right) \cdot {\color{blue}{kx}}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       4. *-commutative N/A

          \[\leadsto \frac{ky}{\sqrt{\left(\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) \cdot {kx}^{2} + 1\right) \cdot {kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{\mathsf{fma}\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, {kx}^{2}, 1\right) \cdot {\color{blue}{kx}}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       6. lower--.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{\mathsf{fma}\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       7. *-commutative N/A

          \[\leadsto \frac{ky}{\sqrt{\mathsf{fma}\left({kx}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{ky}{\sqrt{\mathsf{fma}\left({kx}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       9. unpow2 N/A

          \[\leadsto \frac{ky}{\sqrt{\mathsf{fma}\left(\left(kx \cdot kx\right) \cdot \frac{2}{45} - \frac{1}{3}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{ky}{\sqrt{\mathsf{fma}\left(\left(kx \cdot kx\right) \cdot \frac{2}{45} - \frac{1}{3}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       11. unpow2 N/A

           \[\leadsto \frac{ky}{\sqrt{\mathsf{fma}\left(\left(kx \cdot kx\right) \cdot \frac{2}{45} - \frac{1}{3}, kx \cdot kx, 1\right) \cdot {kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{ky}{\sqrt{\mathsf{fma}\left(\left(kx \cdot kx\right) \cdot \frac{2}{45} - \frac{1}{3}, kx \cdot kx, 1\right) \cdot {kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       13. unpow2 N/A

           \[\leadsto \frac{ky}{\sqrt{\mathsf{fma}\left(\left(kx \cdot kx\right) \cdot \frac{2}{45} - \frac{1}{3}, kx \cdot kx, 1\right) \cdot \left(kx \cdot \color{blue}{kx}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       14. lower-*.f64 24.7

           \[\leadsto \frac{ky}{\sqrt{\mathsf{fma}\left(\left(kx \cdot kx\right) \cdot 0.044444444444444446 - 0.3333333333333333, kx \cdot kx, 1\right) \cdot \left(kx \cdot \color{blue}{kx}\right) + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   11. Applied rewrites 24.7%

       \[\leadsto \frac{ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\left(kx \cdot kx\right) \cdot 0.044444444444444446 - 0.3333333333333333, kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 60.7

         \[\leadsto \sin th \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 42.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{ky}{\sqrt{kx \cdot kx + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.05)
   (*
    (/
     ky
     (sqrt
      (+
       (* kx kx)
       (*
        (fma
         (-
          (*
           (fma (* ky ky) -0.0031746031746031746 0.044444444444444446)
           (* ky ky))
          0.3333333333333333)
         (* ky ky)
         1.0)
        (* ky ky)))))
    (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)))) <= 0.05) {
		tmp = (ky / sqrt(((kx * kx) + (fma(((fma((ky * ky), -0.0031746031746031746, 0.044444444444444446) * (ky * ky)) - 0.3333333333333333), (ky * ky), 1.0) * (ky * ky))))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.05)
		tmp = Float64(Float64(ky / sqrt(Float64(Float64(kx * kx) + Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0031746031746031746, 0.044444444444444446) * Float64(ky * ky)) - 0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(ky / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0031746031746031746 + 0.044444444444444446), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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))))) < 0.050000000000000003

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {ky}^{2}\right) - \frac{1}{3}\right)\right) \cdot \color{blue}{{ky}^{2}}}} \cdot \sin th \]

   5. Applied rewrites 49.4%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\sqrt{\color{blue}{{kx}^{2}} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{ky}{\sqrt{kx \cdot \color{blue}{kx} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{315}, \frac{2}{45}\right) \cdot \left(ky \cdot ky\right) - \frac{1}{3}, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

       2. lower-*.f64 24.6

          \[\leadsto \frac{ky}{\sqrt{kx \cdot \color{blue}{kx} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   11. Applied rewrites 24.6%

       \[\leadsto \frac{ky}{\sqrt{\color{blue}{kx \cdot kx} + \mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0031746031746031746, 0.044444444444444446\right) \cdot \left(ky \cdot ky\right) - 0.3333333333333333, ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}} \cdot \sin th \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 60.7

         \[\leadsto \sin th \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 33.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-56)
   (* (/ (sin ky) (sqrt (* kx kx))) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-56) {
		tmp = (sin(ky) / sqrt((kx * kx))) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-56) then
        tmp = (sin(ky) / sqrt((kx * kx))) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-56) {
		tmp = (Math.sin(ky) / Math.sqrt((kx * kx))) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5e-56:
		tmp = (math.sin(ky) / math.sqrt((kx * kx))) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-56)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(kx * kx))) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-56)
		tmp = (sin(ky) / sqrt((kx * kx))) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-56], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(kx * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.99999999999999997e-56

   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 ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      2. lift-pow.f64 46.2

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \cdot \sin th \]

   5. Applied rewrites 46.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{th} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.3%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{th} \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{kx}^{\color{blue}{2}}}} \cdot th \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx}} \cdot th \]

       2. lower-*.f64 17.2

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx}} \cdot th \]

   11. Applied rewrites 17.2%

       \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx}}} \cdot th \]

   if 4.99999999999999997e-56 < (/.f64 (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 90.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 53.0

         \[\leadsto \sin th \]

   5. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 15.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 2 \cdot 10^{-307}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      2e-307)
   (* (* (* th th) -0.16666666666666666) th)
   th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 2e-307) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)) <= 2d-307) then
        tmp = ((th * th) * (-0.16666666666666666d0)) * th
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th)) <= 2e-307) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)) <= 2e-307:
		tmp = ((th * th) * -0.16666666666666666) * th
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-307)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * th);
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-307)
		tmp = ((th * th) * -0.16666666666666666) * th;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 2e-307], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 1.99999999999999982e-307

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 21.1

         \[\leadsto \sin th \]

   5. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\sin th} \]

   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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 12.3

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   8. Applied rewrites 12.3%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th \]

       2. lower-*.f64 N/A

          \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th \]

       3. pow2 N/A

          \[\leadsto \left(\left(th \cdot th\right) \cdot \frac{-1}{6}\right) \cdot th \]

       4. lift-*.f64 14.1

          \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   11. Applied rewrites 14.1%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   if 1.99999999999999982e-307 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 91.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 22.6

         \[\leadsto \sin th \]

   5. Applied rewrites 22.6%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \]
   7. Step-by-step derivation

   8. Applied rewrites 14.0%

      \[\leadsto th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 30.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 9.2 \cdot 10^{-58}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 9.2e-58)
   (* (* (* th th) -0.16666666666666666) 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)))) <= 9.2e-58) {
		tmp = ((th * th) * -0.16666666666666666) * 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)))) <= 9.2d-58) then
        tmp = ((th * th) * (-0.16666666666666666d0)) * 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)))) <= 9.2e-58) {
		tmp = ((th * th) * -0.16666666666666666) * 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)))) <= 9.2e-58:
		tmp = ((th * th) * -0.16666666666666666) * 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)))) <= 9.2e-58)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * 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)))) <= 9.2e-58)
		tmp = ((th * th) * -0.16666666666666666) * 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], 9.2e-58], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.1999999999999995e-58

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 3.3

         \[\leadsto \sin th \]

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\sin th} \]

   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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 3.4

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   8. Applied rewrites 3.4%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th \]

       2. lower-*.f64 N/A

          \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th \]

       3. pow2 N/A

          \[\leadsto \left(\left(th \cdot th\right) \cdot \frac{-1}{6}\right) \cdot th \]

       4. lift-*.f64 14.3

          \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   11. Applied rewrites 14.3%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   if 9.1999999999999995e-58 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 52.5

         \[\leadsto \sin th \]

   5. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-34}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 1e-34)
   (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
   (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ kx kx)))))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 1e-34) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((kx + kx)))))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(kx), 2.0) <= 1e-34) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((kx + kx)))))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(kx), 2.0) <= 1e-34:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	else:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((kx + kx)))))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 1e-34)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx)))))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(kx) ^ 2.0) <= 1e-34)
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	else
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((kx + kx)))))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 1e-34], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 9.99999999999999928e-35

   1. Initial program 84.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.9

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   if 9.99999999999999928e-35 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      2. lift-pow.f64 53.7

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \cdot \sin th \]

   5. Applied rewrites 53.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      7. cos-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \left(\cos kx \cdot \cos kx - \color{blue}{\sin kx \cdot \sin kx}\right)}} \cdot \sin th \]

      8. cos-sum N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

      10. lower-+.f64 53.2

          \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(kx + kx\right)}} \cdot \sin th \]

   7. Applied rewrites 53.2%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(kx + kx\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 13.6% accurate, 632.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 92.9%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lift-sin.f64 21.7

      \[\leadsto \sin th \]

5. Applied rewrites 21.7%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \]
7. Step-by-step derivation

8. Applied rewrites 13.2%

   \[\leadsto th \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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)))