Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.7%

Time: 8.5s

Alternatives: 23

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: 94.2% 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 94.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.7

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

3. Applied rewrites 99.7%

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

Alternative 2: 80.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;t\_2 \leq -0.985:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right) \cdot \sin ky}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\frac{t\_3}{\mathsf{hypot}\left(t\_3, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.96:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(th \cdot th\right) - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right) \cdot \sin ky}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= t_2 -0.985)
     (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
     (if (<= t_2 -0.1)
       (/
        (*
         (*
          (fma
           (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
           (* th th)
           1.0)
          th)
         (sin ky))
        t_1)
       (if (<= t_2 0.01)
         (* (/ t_3 (hypot t_3 (sin kx))) (sin th))
         (if (<= t_2 0.96)
           (/
            (*
             (*
              (fma
               (-
                (*
                 (fma (* th th) -0.0001984126984126984 0.008333333333333333)
                 (* th th))
                0.16666666666666666)
               (* th th)
               1.0)
              th)
             (sin ky))
            t_1)
           (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (t_2 <= -0.985) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = ((fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th) * sin(ky)) / t_1;
	} else if (t_2 <= 0.01) {
		tmp = (t_3 / hypot(t_3, sin(kx))) * sin(th);
	} else if (t_2 <= 0.96) {
		tmp = ((fma(((fma((th * th), -0.0001984126984126984, 0.008333333333333333) * (th * th)) - 0.16666666666666666), (th * th), 1.0) * th) * sin(ky)) / t_1;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (t_2 <= -0.985)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th) * sin(ky)) / t_1);
	elseif (t_2 <= 0.01)
		tmp = Float64(Float64(t_3 / hypot(t_3, sin(kx))) * sin(th));
	elseif (t_2 <= 0.96)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(th * th), -0.0001984126984126984, 0.008333333333333333) * Float64(th * th)) - 0.16666666666666666), Float64(th * th), 1.0) * th) * sin(ky)) / t_1);
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[t$95$2, -0.985], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.01], N[(N[(t$95$3 / N[Sqrt[t$95$3 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.96], N[(N[(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] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 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))))) < -0.984999999999999987

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 57.9%

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

   if -0.984999999999999987 < (/.f64 (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 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 47.2

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

   6. Applied rewrites 47.2%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   9. Applied rewrites 55.1%

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

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   6. Applied rewrites 47.2%

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

   if 0.95999999999999996 < (/.f64 (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 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.5%

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

Alternative 3: 80.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right) \cdot \sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ t_3 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;t\_1 \leq -0.985:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\frac{t\_3}{\mathsf{hypot}\left(t\_3, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.96:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2
         (/
          (*
           (*
            (fma
             (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
             (* th th)
             1.0)
            th)
           (sin ky))
          (hypot (sin kx) (sin ky))))
        (t_3 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= t_1 -0.985)
     (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
     (if (<= t_1 -0.1)
       t_2
       (if (<= t_1 0.01)
         (* (/ t_3 (hypot t_3 (sin kx))) (sin th))
         (if (<= t_1 0.96) t_2 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = ((fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th) * sin(ky)) / hypot(sin(kx), sin(ky));
	double t_3 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (t_1 <= -0.985) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = t_2;
	} else if (t_1 <= 0.01) {
		tmp = (t_3 / hypot(t_3, sin(kx))) * sin(th);
	} else if (t_1 <= 0.96) {
		tmp = t_2;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th) * sin(ky)) / hypot(sin(kx), sin(ky)))
	t_3 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (t_1 <= -0.985)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = t_2;
	elseif (t_1 <= 0.01)
		tmp = Float64(Float64(t_3 / hypot(t_3, sin(kx))) * sin(th));
	elseif (t_1 <= 0.96)
		tmp = t_2;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[t$95$1, -0.985], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], t$95$2, If[LessEqual[t$95$1, 0.01], N[(N[(t$95$3 / N[Sqrt[t$95$3 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.96], t$95$2, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.984999999999999987

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 57.9%

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

   if -0.984999999999999987 < (/.f64 (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 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 47.2

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

   6. Applied rewrites 47.2%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   9. Applied rewrites 55.1%

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

   if 0.95999999999999996 < (/.f64 (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 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.5%

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

Alternative 4: 80.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\\ t_3 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;t\_1 \leq -0.985:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\frac{t\_3}{\mathsf{hypot}\left(t\_3, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.96:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2
         (*
          (/ (sin ky) (hypot (sin ky) (sin kx)))
          (*
           (fma
            (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
            (* th th)
            1.0)
           th)))
        (t_3 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= t_1 -0.985)
     (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
     (if (<= t_1 -0.1)
       t_2
       (if (<= t_1 0.01)
         (* (/ t_3 (hypot t_3 (sin kx))) (sin th))
         (if (<= t_1 0.96) t_2 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (sin(ky) / hypot(sin(ky), sin(kx))) * (fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th);
	double t_3 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (t_1 <= -0.985) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = t_2;
	} else if (t_1 <= 0.01) {
		tmp = (t_3 / hypot(t_3, sin(kx))) * sin(th);
	} else if (t_1 <= 0.96) {
		tmp = t_2;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th))
	t_3 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (t_1 <= -0.985)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = t_2;
	elseif (t_1 <= 0.01)
		tmp = Float64(Float64(t_3 / hypot(t_3, sin(kx))) * sin(th));
	elseif (t_1 <= 0.96)
		tmp = t_2;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] / 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]}, Block[{t$95$3 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[t$95$1, -0.985], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], t$95$2, If[LessEqual[t$95$1, 0.01], N[(N[(t$95$3 / N[Sqrt[t$95$3 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.96], t$95$2, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.984999999999999987

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 57.9%

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

   if -0.984999999999999987 < (/.f64 (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 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)} \]
   5. 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 50.9

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

   6. Applied rewrites 50.9%

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

   if -0.10000000000000001 < (/.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))))) < 0.0100000000000000002

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   9. Applied rewrites 55.1%

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

   if 0.95999999999999996 < (/.f64 (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 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.5%

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

Alternative 5: 80.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;t\_2 \leq -0.985:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\frac{t\_3}{\mathsf{hypot}\left(t\_3, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.96:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (/ (sin ky) (hypot (sin ky) (sin kx)))
          (* (fma (* th th) -0.16666666666666666 1.0) th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= t_2 -0.985)
     (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
     (if (<= t_2 -0.1)
       t_1
       (if (<= t_2 0.01)
         (* (/ t_3 (hypot t_3 (sin kx))) (sin th))
         (if (<= t_2 0.96) t_1 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), sin(kx))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (t_2 <= -0.985) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = t_1;
	} else if (t_2 <= 0.01) {
		tmp = (t_3 / hypot(t_3, sin(kx))) * sin(th);
	} else if (t_2 <= 0.96) {
		tmp = t_1;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (t_2 <= -0.985)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = t_1;
	elseif (t_2 <= 0.01)
		tmp = Float64(Float64(t_3 / hypot(t_3, sin(kx))) * sin(th));
	elseif (t_2 <= 0.96)
		tmp = t_1;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[t$95$2, -0.985], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], t$95$1, If[LessEqual[t$95$2, 0.01], N[(N[(t$95$3 / N[Sqrt[t$95$3 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.96], t$95$1, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.984999999999999987

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 57.9%

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

   if -0.984999999999999987 < (/.f64 (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 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.9

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

   6. Applied rewrites 50.9%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   9. Applied rewrites 55.1%

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

   if 0.95999999999999996 < (/.f64 (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 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.5%

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

Alternative 6: 80.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ \mathbf{if}\;t\_2 \leq -0.985:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_3}\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.96:\\ \;\;\;\;\sin ky \cdot \left(th \cdot \frac{1}{t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (hypot (sin kx) (sin ky))))
   (if (<= t_2 -0.985)
     (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
     (if (<= t_2 -0.1)
       (/ (* (sin ky) th) t_3)
       (if (<= t_2 0.01)
         (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
         (if (<= t_2 0.96)
           (* (sin ky) (* th (/ 1.0 t_3)))
           (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = hypot(sin(kx), sin(ky));
	double tmp;
	if (t_2 <= -0.985) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (sin(ky) * th) / t_3;
	} else if (t_2 <= 0.01) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else if (t_2 <= 0.96) {
		tmp = sin(ky) * (th * (1.0 / t_3));
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = hypot(sin(kx), sin(ky))
	tmp = 0.0
	if (t_2 <= -0.985)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(sin(ky) * th) / t_3);
	elseif (t_2 <= 0.01)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	elseif (t_2 <= 0.96)
		tmp = Float64(sin(ky) * Float64(th * Float64(1.0 / t_3)));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$2, -0.985], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 0.01], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.96], N[(N[Sin[ky], $MachinePrecision] * N[(th * N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 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))))) < -0.984999999999999987

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 57.9%

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

   if -0.984999999999999987 < (/.f64 (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 94.2%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. pow2 N/A

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

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

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

   6. Applied rewrites 47.4%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   9. Applied rewrites 55.1%

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

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   1. Initial program 94.2%

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

   2. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   3. 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-hypot.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. sqrt-div N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 51.1%

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

   if 0.95999999999999996 < (/.f64 (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 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.5%

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

Alternative 7: 80.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_2 \leq -0.985:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.96:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))))
   (if (<= t_2 -0.985)
     (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
     (if (<= t_2 -0.1)
       t_3
       (if (<= t_2 0.01)
         (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
         (if (<= t_2 0.96) t_3 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double tmp;
	if (t_2 <= -0.985) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = t_3;
	} else if (t_2 <= 0.01) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else if (t_2 <= 0.96) {
		tmp = t_3;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	tmp = 0.0
	if (t_2 <= -0.985)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = t_3;
	elseif (t_2 <= 0.01)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	elseif (t_2 <= 0.96)
		tmp = t_3;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.985], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], t$95$3, If[LessEqual[t$95$2, 0.01], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.96], t$95$3, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.984999999999999987

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 57.9%

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

   if -0.984999999999999987 < (/.f64 (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 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   1. Initial program 94.2%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. pow2 N/A

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

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

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

   6. Applied rewrites 47.4%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   9. Applied rewrites 55.1%

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

   if 0.95999999999999996 < (/.f64 (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 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.5%

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

Alternative 8: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_2 \leq -0.985:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.96:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))))
   (if (<= t_2 -0.985)
     (/ (* (sin th) (sin ky)) (hypot kx (sin ky)))
     (if (<= t_2 -0.1)
       t_3
       (if (<= t_2 0.01)
         (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
         (if (<= t_2 0.96) t_3 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double tmp;
	if (t_2 <= -0.985) {
		tmp = (sin(th) * sin(ky)) / hypot(kx, sin(ky));
	} else if (t_2 <= -0.1) {
		tmp = t_3;
	} else if (t_2 <= 0.01) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else if (t_2 <= 0.96) {
		tmp = t_3;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	tmp = 0.0
	if (t_2 <= -0.985)
		tmp = Float64(Float64(sin(th) * sin(ky)) / hypot(kx, sin(ky)));
	elseif (t_2 <= -0.1)
		tmp = t_3;
	elseif (t_2 <= 0.01)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	elseif (t_2 <= 0.96)
		tmp = t_3;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.985], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], t$95$3, If[LessEqual[t$95$2, 0.01], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.96], t$95$3, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.984999999999999987

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in kx around 0

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

      1. remove-double-div 54.1

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

   6. Applied rewrites 54.1%

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

   if -0.984999999999999987 < (/.f64 (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 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   1. Initial program 94.2%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. pow2 N/A

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

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

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

   6. Applied rewrites 47.4%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   9. Applied rewrites 55.1%

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

   if 0.95999999999999996 < (/.f64 (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 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.5%

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

Alternative 9: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;t\_2 \leq -0.985:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\frac{t\_3}{\mathsf{hypot}\left(t\_3, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.96:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin kx) (sin ky))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= t_2 -0.985)
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) (sin th))
     (if (<= t_2 -0.1)
       t_1
       (if (<= t_2 0.01)
         (* (/ t_3 (hypot t_3 (sin kx))) (sin th))
         (if (<= t_2 0.96) t_1 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (t_2 <= -0.985) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = t_1;
	} else if (t_2 <= 0.01) {
		tmp = (t_3 / hypot(t_3, sin(kx))) * sin(th);
	} else if (t_2 <= 0.96) {
		tmp = t_1;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (t_2 <= -0.985)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = t_1;
	elseif (t_2 <= 0.01)
		tmp = Float64(Float64(t_3 / hypot(t_3, sin(kx))) * sin(th));
	elseif (t_2 <= 0.96)
		tmp = t_1;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[t$95$2, -0.985], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], t$95$1, If[LessEqual[t$95$2, 0.01], N[(N[(t$95$3 / N[Sqrt[t$95$3 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.96], t$95$1, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.984999999999999987

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 43.5

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

   4. Applied rewrites 43.5%

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

   5. Taylor expanded in kx around 0

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

      1. pow2 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 31.0

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

   7. Applied rewrites 31.0%

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

   if -0.984999999999999987 < (/.f64 (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 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   1. Initial program 94.2%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. pow2 N/A

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

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

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

   6. Applied rewrites 47.4%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.1

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

   9. Applied rewrites 55.1%

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

   if 0.95999999999999996 < (/.f64 (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 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.5%

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

Alternative 10: 67.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;ky \leq 190:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= ky 190.0)
     (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (ky <= 190.0) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (ky <= 190.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[ky, 190.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 190

   1. Initial program 94.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      7. lower-*.f64 55.1

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   9. Applied rewrites 55.1%

      \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   if 190 < ky

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, \color{blue}{kx}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      8. lower-*.f64 43.5

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   4. Applied rewrites 43.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(kx, kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \color{blue}{\frac{1}{2}}}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \color{blue}{\frac{1}{2}}}} \cdot \sin th \]

      6. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(ky + ky\right) \cdot \frac{1}{2}}} \cdot \sin th \]

      8. lower-+.f64 31.0

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th \]

   7. Applied rewrites 31.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \cos \left(ky + ky\right) \cdot 0.5}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) -0.1)
   (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) th)
   (* (/ ky (hypot ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= -0.1) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= -0.1) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (0.5 * Math.cos((ky + ky)))))) * th;
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= -0.1:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (0.5 * math.cos((ky + ky)))))) * th
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.1)
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot th \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx} + {\sin ky}^{2}}} \cdot th \]

      4. lower-*.f64 30.7

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx} + {\sin ky}^{2}}} \cdot th \]

   7. Applied rewrites 30.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot th \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot th \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]

      4. cos-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \left(\cos ky \cdot \cos ky - \color{blue}{\sin ky \cdot \sin ky}\right)}} \cdot th \]

      5. cos-sum N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(ky + ky\right)}}} \cdot th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot th \]

      8. lower-+.f64 16.7

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th \]

   10. Applied rewrites 16.7%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}} \cdot th \]

   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)))))

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]
   5. Step-by-step derivation

   6. Applied rewrites 51.3%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 65.5%

      \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.9973733552028557:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) th)
     (if (<= t_1 0.01)
       (*
        (/ (* (fma (* ky ky) -0.16666666666666666 1.0) ky) (sin kx))
        (sin th))
       (if (<= t_1 0.9973733552028557)
         (sin th)
         (*
          (/
           ky
           (hypot
            ky
            (*
             (fma
              (-
               (*
                (fma -0.0001984126984126984 (* kx kx) 0.008333333333333333)
                (* kx kx))
               0.16666666666666666)
              (* kx kx)
              1.0)
             kx)))
          (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	} else if (t_1 <= 0.01) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 0.9973733552028557) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot(ky, (fma(((fma(-0.0001984126984126984, (kx * kx), 0.008333333333333333) * (kx * kx)) - 0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * th);
	elseif (t_1 <= 0.01)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 0.9973733552028557)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(ky, Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(kx * kx), 0.008333333333333333) * Float64(kx * kx)) - 0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9973733552028557], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(kx * kx), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot th \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx} + {\sin ky}^{2}}} \cdot th \]

      4. lower-*.f64 30.7

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx} + {\sin ky}^{2}}} \cdot th \]

   7. Applied rewrites 30.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot th \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot th \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]

      4. cos-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \left(\cos ky \cdot \cos ky - \color{blue}{\sin ky \cdot \sin ky}\right)}} \cdot th \]

      5. cos-sum N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(ky + ky\right)}}} \cdot th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot th \]

      8. lower-+.f64 16.7

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th \]

   10. Applied rewrites 16.7%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}} \cdot th \]

   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))))) < 0.0100000000000000002

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. lower-*.f64 84.7

         \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

   3. Applied rewrites 84.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   4. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      5. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \sin th \]

      12. pow1/2 N/A

          \[\leadsto \frac{\sin ky}{{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}^{\color{blue}{\frac{1}{2}}}} \cdot \sin th \]

      13. sqr-sin-a-rev N/A

          \[\leadsto \frac{\sin ky}{{\left(\sin kx \cdot \sin kx\right)}^{\frac{1}{2}}} \cdot \sin th \]

      14. pow2 N/A

          \[\leadsto \frac{\sin ky}{{\left({\sin kx}^{2}\right)}^{\frac{1}{2}}} \cdot \sin th \]

      15. pow-pow N/A

          \[\leadsto \frac{\sin ky}{{\sin kx}^{\color{blue}{\left(2 \cdot \frac{1}{2}\right)}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{{\sin kx}^{1}} \cdot \sin th \]

      17. unpow1 N/A

          \[\leadsto \frac{\sin ky}{\sin kx} \cdot \sin th \]

   6. Applied rewrites 27.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sin kx} \cdot \sin th \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sin kx} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sin kx} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky}{\sin kx} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky}{\sin kx} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky}{\sin kx} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sin kx} \cdot \sin th \]

      7. lower-*.f64 24.6

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sin kx} \cdot \sin th \]

   9. Applied rewrites 24.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\sin kx} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.997373355202855749

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 23.4

         \[\leadsto \sin th \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\sin th} \]

   if 0.997373355202855749 < (/.f64 (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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

      2. unpow1 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{{\sin kx}^{1}}\right)} \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, {\sin kx}^{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}}\right)} \cdot \sin th \]

      4. pow-neg N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\frac{1}{{\sin kx}^{-1}}}\right)} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\sin kx}^{\color{blue}{\left(-2 \cdot \frac{1}{2}\right)}}}\right)} \cdot \sin th \]

      6. pow-pow N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{{\left({\sin kx}^{-2}\right)}^{\frac{1}{2}}}}\right)} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left({\sin kx}^{\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      8. pow-flip N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\color{blue}{\left(\frac{1}{{\sin kx}^{2}}\right)}}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      9. pow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{\sin kx \cdot \sin kx}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      10. sqr-sin-a-rev N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      11. pow1/2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}}\right)} \cdot \sin th \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\frac{1}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}}\right)} \cdot \sin th \]

      13. pow1/2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{{\left(\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}^{\frac{1}{2}}}}\right)} \cdot \sin th \]

      14. sqr-sin-a-rev N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{\sin kx \cdot \sin kx}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      15. pow2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{{\sin kx}^{2}}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      16. pow-flip N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\color{blue}{\left({\sin kx}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      17. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left({\sin kx}^{\color{blue}{-2}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      18. pow-pow N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{{\sin kx}^{\left(-2 \cdot \frac{1}{2}\right)}}}\right)} \cdot \sin th \]

      19. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\sin kx}^{\color{blue}{-1}}}\right)} \cdot \sin th \]

      20. inv-pow N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{\frac{1}{\sin kx}}}\right)} \cdot \sin th \]

      21. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{\frac{1}{\sin kx}}}\right)} \cdot \sin th \]

      22. lift-sin.f64 99.6

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{1}{\color{blue}{\sin kx}}}\right)} \cdot \sin th \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\frac{1}{\frac{1}{\sin kx}}}\right)} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right)}\right)} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

   8. Applied rewrites 57.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, kx \cdot kx, \frac{1}{120}\right) \cdot \left(kx \cdot kx\right) - \frac{1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 33.7%

       \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

   12. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, kx \cdot kx, \frac{1}{120}\right) \cdot \left(kx \cdot kx\right) - \frac{1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
   13. Step-by-step derivation

   14. Applied rewrites 45.9%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 46.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.9973733552028557:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) th)
     (if (<= t_1 0.01)
       (* (/ ky (sin kx)) (sin th))
       (if (<= t_1 0.9973733552028557)
         (sin th)
         (*
          (/
           ky
           (hypot
            ky
            (*
             (fma
              (-
               (*
                (fma -0.0001984126984126984 (* kx kx) 0.008333333333333333)
                (* kx kx))
               0.16666666666666666)
              (* kx kx)
              1.0)
             kx)))
          (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	} else if (t_1 <= 0.01) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_1 <= 0.9973733552028557) {
		tmp = sin(th);
	} else {
		tmp = (ky / hypot(ky, (fma(((fma(-0.0001984126984126984, (kx * kx), 0.008333333333333333) * (kx * kx)) - 0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * th);
	elseif (t_1 <= 0.01)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_1 <= 0.9973733552028557)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / hypot(ky, Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(kx * kx), 0.008333333333333333) * Float64(kx * kx)) - 0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9973733552028557], N[Sin[th], $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(kx * kx), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot th \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx} + {\sin ky}^{2}}} \cdot th \]

      4. lower-*.f64 30.7

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx} + {\sin ky}^{2}}} \cdot th \]

   7. Applied rewrites 30.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot th \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot th \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot th \]

      4. cos-2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \left(\cos ky \cdot \cos ky - \color{blue}{\sin ky \cdot \sin ky}\right)}} \cdot th \]

      5. cos-sum N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(ky + ky\right)}}} \cdot th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot th \]

      8. lower-+.f64 16.7

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th \]

   10. Applied rewrites 16.7%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}} \cdot th \]

   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))))) < 0.0100000000000000002

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 24.9

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.997373355202855749

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 23.4

         \[\leadsto \sin th \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\sin th} \]

   if 0.997373355202855749 < (/.f64 (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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

      2. unpow1 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{{\sin kx}^{1}}\right)} \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, {\sin kx}^{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}}\right)} \cdot \sin th \]

      4. pow-neg N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\frac{1}{{\sin kx}^{-1}}}\right)} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\sin kx}^{\color{blue}{\left(-2 \cdot \frac{1}{2}\right)}}}\right)} \cdot \sin th \]

      6. pow-pow N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{{\left({\sin kx}^{-2}\right)}^{\frac{1}{2}}}}\right)} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left({\sin kx}^{\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      8. pow-flip N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\color{blue}{\left(\frac{1}{{\sin kx}^{2}}\right)}}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      9. pow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{\sin kx \cdot \sin kx}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      10. sqr-sin-a-rev N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      11. pow1/2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}}\right)} \cdot \sin th \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\frac{1}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}}\right)} \cdot \sin th \]

      13. pow1/2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{{\left(\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}^{\frac{1}{2}}}}\right)} \cdot \sin th \]

      14. sqr-sin-a-rev N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{\sin kx \cdot \sin kx}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      15. pow2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{{\sin kx}^{2}}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      16. pow-flip N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\color{blue}{\left({\sin kx}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      17. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left({\sin kx}^{\color{blue}{-2}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      18. pow-pow N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{{\sin kx}^{\left(-2 \cdot \frac{1}{2}\right)}}}\right)} \cdot \sin th \]

      19. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\sin kx}^{\color{blue}{-1}}}\right)} \cdot \sin th \]

      20. inv-pow N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{\frac{1}{\sin kx}}}\right)} \cdot \sin th \]

      21. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{\frac{1}{\sin kx}}}\right)} \cdot \sin th \]

      22. lift-sin.f64 99.6

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{1}{\color{blue}{\sin kx}}}\right)} \cdot \sin th \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\frac{1}{\frac{1}{\sin kx}}}\right)} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right)}\right)} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

   8. Applied rewrites 57.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, kx \cdot kx, \frac{1}{120}\right) \cdot \left(kx \cdot kx\right) - \frac{1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 33.7%

       \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

   12. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, kx \cdot kx, \frac{1}{120}\right) \cdot \left(kx \cdot kx\right) - \frac{1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
   13. Step-by-step derivation

   14. Applied rewrites 45.9%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 46.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.1:\\ \;\;\;\;\left(ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.1)
   (* (* ky th) (/ 1.0 (hypot (sin kx) ky)))
   (if (<= (sin kx) 5e-10)
     (*
      (/
       ky
       (hypot
        ky
        (*
         (fma
          (-
           (*
            (fma -0.0001984126984126984 (* kx kx) 0.008333333333333333)
            (* kx kx))
           0.16666666666666666)
          (* kx kx)
          1.0)
         kx)))
      (sin th))
     (* (/ ky (sin kx)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.1) {
		tmp = (ky * th) * (1.0 / hypot(sin(kx), ky));
	} else if (sin(kx) <= 5e-10) {
		tmp = (ky / hypot(ky, (fma(((fma(-0.0001984126984126984, (kx * kx), 0.008333333333333333) * (kx * kx)) - 0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = (ky / sin(kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.1)
		tmp = Float64(Float64(ky * th) * Float64(1.0 / hypot(sin(kx), ky)));
	elseif (sin(kx) <= 5e-10)
		tmp = Float64(Float64(ky / hypot(ky, Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(kx * kx), 0.008333333333333333) * Float64(kx * kx)) - 0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.1], N[(N[(ky * th), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 5e-10], N[(N[(ky / N[Sqrt[ky ^ 2 + N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(kx * kx), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.10000000000000001

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   3. 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. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{\color{blue}{1}}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      5. sqrt-div N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}} \]

      9. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \]

      10. lower-hypot.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \color{blue}{\sin ky}\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin \color{blue}{ky}\right)} \]

      12. lift-sin.f64 47.4

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.1%

      \[\leadsto \left(ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, ky\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 30.3%

       \[\leadsto \left(ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, ky\right)} \]

   if -0.10000000000000001 < (sin.f64 kx) < 5.00000000000000031e-10

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

      2. unpow1 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{{\sin kx}^{1}}\right)} \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, {\sin kx}^{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}}\right)} \cdot \sin th \]

      4. pow-neg N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\frac{1}{{\sin kx}^{-1}}}\right)} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\sin kx}^{\color{blue}{\left(-2 \cdot \frac{1}{2}\right)}}}\right)} \cdot \sin th \]

      6. pow-pow N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{{\left({\sin kx}^{-2}\right)}^{\frac{1}{2}}}}\right)} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left({\sin kx}^{\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      8. pow-flip N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\color{blue}{\left(\frac{1}{{\sin kx}^{2}}\right)}}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      9. pow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{\sin kx \cdot \sin kx}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      10. sqr-sin-a-rev N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      11. pow1/2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}}\right)} \cdot \sin th \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\frac{1}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}}}\right)} \cdot \sin th \]

      13. pow1/2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{{\left(\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}\right)}^{\frac{1}{2}}}}\right)} \cdot \sin th \]

      14. sqr-sin-a-rev N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{\sin kx \cdot \sin kx}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      15. pow2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left(\frac{1}{\color{blue}{{\sin kx}^{2}}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      16. pow-flip N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\color{blue}{\left({\sin kx}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      17. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\left({\sin kx}^{\color{blue}{-2}}\right)}^{\frac{1}{2}}}\right)} \cdot \sin th \]

      18. pow-pow N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{{\sin kx}^{\left(-2 \cdot \frac{1}{2}\right)}}}\right)} \cdot \sin th \]

      19. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{{\sin kx}^{\color{blue}{-1}}}\right)} \cdot \sin th \]

      20. inv-pow N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{\frac{1}{\sin kx}}}\right)} \cdot \sin th \]

      21. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\color{blue}{\frac{1}{\sin kx}}}\right)} \cdot \sin th \]

      22. lift-sin.f64 99.6

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{1}{\color{blue}{\sin kx}}}\right)} \cdot \sin th \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\frac{1}{\frac{1}{\sin kx}}}\right)} \cdot \sin th \]

   6. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right)}\right)} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

   8. Applied rewrites 57.6%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, kx \cdot kx, \frac{1}{120}\right) \cdot \left(kx \cdot kx\right) - \frac{1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 33.7%

       \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

   12. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, kx \cdot kx, \frac{1}{120}\right) \cdot \left(kx \cdot kx\right) - \frac{1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
   13. Step-by-step derivation

   14. Applied rewrites 45.9%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

   if 5.00000000000000031e-10 < (sin.f64 kx)

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 24.9

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 46.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;\left(t\_1 \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, t\_1\right)}\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.1)
     (* (* t_1 th) (/ 1.0 (hypot (sin kx) t_1)))
     (if (<= t_2 0.01) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = (t_1 * th) * (1.0 / hypot(sin(kx), t_1));
	} else if (t_2 <= 0.01) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = Float64(Float64(t_1 * th) * Float64(1.0 / hypot(sin(kx), t_1)));
	elseif (t_2 <= 0.01)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.1], N[(N[(t$95$1 * th), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.01], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   3. 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. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{\color{blue}{1}}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      5. sqrt-div N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}} \]

      9. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \]

      10. lower-hypot.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \color{blue}{\sin ky}\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin \color{blue}{ky}\right)} \]

      12. lift-sin.f64 47.4

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

      7. lower-*.f64 22.8

         \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   7. Applied rewrites 22.8%

      \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, ky \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky\right)} \]

      7. lower-*.f64 24.6

         \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)} \]

   10. Applied rewrites 24.6%

       \[\leadsto \left(\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot \color{blue}{ky}\right)} \]

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 24.9

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 23.4

         \[\leadsto \sin th \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.01)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.01) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.01d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.01) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.01:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 24.9

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   4. Applied rewrites 24.9%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 23.4

         \[\leadsto \sin th \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 44.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.01)
   (/ (* (sin th) ky) (sin kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.01) {
		tmp = (sin(th) * ky) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.01d0) then
        tmp = (sin(th) * ky) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.01) {
		tmp = (Math.sin(th) * ky) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.01:
		tmp = (math.sin(th) * ky) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = (sin(th) * ky) / sin(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sin kx}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin \color{blue}{kx}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin \color{blue}{kx}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      5. lift-sin.f64 24.1

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   4. Applied rewrites 24.1%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   if 0.0100000000000000002 < (/.f64 (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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 23.4

         \[\leadsto \sin th \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 43.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0001:\\ \;\;\;\;\left(ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.0001)
   (* (* ky th) (/ 1.0 (hypot (sin kx) ky)))
   (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)))) <= 0.0001) {
		tmp = (ky * th) * (1.0 / hypot(sin(kx), ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.0001) {
		tmp = (ky * th) * (1.0 / Math.hypot(Math.sin(kx), ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.0001:
		tmp = (ky * th) * (1.0 / math.hypot(math.sin(kx), ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = Float64(Float64(ky * th) * Float64(1.0 / hypot(sin(kx), ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.0001)
		tmp = (ky * th) * (1.0 / hypot(sin(kx), ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(ky * th), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $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))))) < 1.00000000000000005e-4

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   3. 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. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{\color{blue}{1}}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      5. sqrt-div N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}} \]

      9. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \]

      10. lower-hypot.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \color{blue}{\sin ky}\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin \color{blue}{ky}\right)} \]

      12. lift-sin.f64 47.4

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \left(ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.1%

      \[\leadsto \left(ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \left(ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, ky\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 30.3%

       \[\leadsto \left(ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, ky\right)} \]

   if 1.00000000000000005e-4 < (/.f64 (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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 23.4

         \[\leadsto \sin th \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 37.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-5)
   (* ky (/ th (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-5) {
		tmp = ky * (th / 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-5) then
        tmp = ky * (th / 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-5) {
		tmp = ky * (th / 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-5:
		tmp = ky * (th / 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-5)
		tmp = Float64(ky * Float64(th / 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-5)
		tmp = ky * (th / 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-5], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $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))))) < 5.00000000000000024e-5

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   3. 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. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{\color{blue}{1}}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      5. sqrt-div N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}} \]

      9. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \]

      10. lower-hypot.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \color{blue}{\sin ky}\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin \color{blue}{ky}\right)} \]

      12. lift-sin.f64 47.4

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\sin kx}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      2. lower-*.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sin kx}} \]

      3. lower-/.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

      4. lift-sin.f64 15.9

         \[\leadsto ky \cdot \frac{th}{\sin kx} \]

   7. Applied rewrites 15.9%

      \[\leadsto ky \cdot \color{blue}{\frac{th}{\sin kx}} \]

   if 5.00000000000000024e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 23.4

         \[\leadsto \sin th \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 35.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{ky}{\sqrt{kx \cdot kx + {ky}^{2}}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-48)
   (* (/ ky (sqrt (+ (* kx kx) (pow ky 2.0)))) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 4e-48) {
		tmp = (ky / sqrt(((kx * kx) + pow(ky, 2.0)))) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 4d-48) then
        tmp = (ky / sqrt(((kx * kx) + (ky ** 2.0d0)))) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 4e-48) {
		tmp = (ky / Math.sqrt(((kx * kx) + Math.pow(ky, 2.0)))) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 4e-48:
		tmp = (ky / math.sqrt(((kx * kx) + math.pow(ky, 2.0)))) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-48)
		tmp = Float64(Float64(ky / sqrt(Float64(Float64(kx * kx) + (ky ^ 2.0)))) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-48)
		tmp = (ky / sqrt(((kx * kx) + (ky ^ 2.0)))) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-48], N[(N[(ky / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[Power[ky, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.9999999999999999e-48

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot th \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx} + {\sin ky}^{2}}} \cdot th \]

      4. lower-*.f64 30.7

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx} + {\sin ky}^{2}}} \cdot th \]

   7. Applied rewrites 30.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot th \]
   9. Step-by-step derivation

   10. Applied rewrites 18.5%

       \[\leadsto \frac{\color{blue}{ky}}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot th \]

   11. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\sqrt{kx \cdot kx + {\color{blue}{ky}}^{2}}} \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 22.1%

       \[\leadsto \frac{ky}{\sqrt{kx \cdot kx + {\color{blue}{ky}}^{2}}} \cdot th \]

   if 3.9999999999999999e-48 < (/.f64 (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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 23.4

         \[\leadsto \sin th \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 27.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 4 \cdot 10^{-34}:\\ \;\;\;\;\frac{ky}{\sqrt{kx \cdot kx + {ky}^{2}}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 4e-34)
   (* (/ ky (sqrt (+ (* kx kx) (pow ky 2.0)))) 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)))) <= 4e-34) {
		tmp = (ky / sqrt(((kx * kx) + pow(ky, 2.0)))) * 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)))) <= 4d-34) then
        tmp = (ky / sqrt(((kx * kx) + (ky ** 2.0d0)))) * 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)))) <= 4e-34) {
		tmp = (ky / Math.sqrt(((kx * kx) + Math.pow(ky, 2.0)))) * 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)))) <= 4e-34:
		tmp = (ky / math.sqrt(((kx * kx) + math.pow(ky, 2.0)))) * th
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 4e-34)
		tmp = Float64(Float64(ky / sqrt(Float64(Float64(kx * kx) + (ky ^ 2.0)))) * 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)))) <= 4e-34)
		tmp = (ky / sqrt(((kx * kx) + (ky ^ 2.0)))) * th;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-34], N[(N[(ky / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[Power[ky, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.99999999999999971e-34

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{th} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot th \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

      2. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot th \]

      3. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx} + {\sin ky}^{2}}} \cdot th \]

      4. lower-*.f64 30.7

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot \color{blue}{kx} + {\sin ky}^{2}}} \cdot th \]

   7. Applied rewrites 30.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot th \]
   9. Step-by-step derivation

   10. Applied rewrites 18.5%

       \[\leadsto \frac{\color{blue}{ky}}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot th \]

   11. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\sqrt{kx \cdot kx + {\color{blue}{ky}}^{2}}} \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 22.1%

       \[\leadsto \frac{ky}{\sqrt{kx \cdot kx + {\color{blue}{ky}}^{2}}} \cdot th \]

   if 3.99999999999999971e-34 < (/.f64 (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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   3. 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. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{\color{blue}{1}}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      5. sqrt-div N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}} \]

      9. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \]

      10. lower-hypot.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \color{blue}{\sin ky}\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin \color{blue}{ky}\right)} \]

      12. lift-sin.f64 47.4

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto th \]
   6. Step-by-step derivation

   7. Applied rewrites 13.3%

      \[\leadsto th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 15.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 3 \cdot 10^{-299}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      3e-299)
   (* (* (* th th) th) -0.16666666666666666)
   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)) <= 3e-299) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} 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)) <= 3d-299) then
        tmp = ((th * th) * th) * (-0.16666666666666666d0)
    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)) <= 3e-299) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} 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)) <= 3e-299:
		tmp = ((th * th) * th) * -0.16666666666666666
	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)) <= 3e-299)
		tmp = Float64(Float64(Float64(th * th) * th) * -0.16666666666666666);
	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)) <= 3e-299)
		tmp = ((th * th) * th) * -0.16666666666666666;
	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], 3e-299], N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666), $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)) < 2.99999999999999984e-299

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 23.4

         \[\leadsto \sin th \]

   4. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\sin th} \]

   5. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 13.0

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   7. Applied rewrites 13.0%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   8. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

      3. unpow3 N/A

         \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

      4. pow2 N/A

         \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

      6. pow2 N/A

         \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

      7. lift-*.f64 10.8

         \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   10. Applied rewrites 10.8%

       \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   if 2.99999999999999984e-299 < (*.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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   3. 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. *-commutative N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{\color{blue}{1}}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      5. sqrt-div N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}} \]

      9. unpow2 N/A

         \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \]

      10. lower-hypot.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \color{blue}{\sin ky}\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin \color{blue}{ky}\right)} \]

      12. lift-sin.f64 47.4

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto th \]
   6. Step-by-step derivation

   7. Applied rewrites 13.3%

      \[\leadsto th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 13.3% accurate, 170.4× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 94.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
3. 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. *-commutative N/A

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   4. lift-sin.f64 N/A

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \sqrt{\frac{\color{blue}{1}}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

   5. sqrt-div N/A

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   6. metadata-eval N/A

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   8. unpow2 N/A

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + {\sin ky}^{2}}} \]

   9. unpow2 N/A

      \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}} \]

   10. lower-hypot.f64 N/A

       \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \color{blue}{\sin ky}\right)} \]

   11. lift-sin.f64 N/A

       \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin \color{blue}{ky}\right)} \]

   12. lift-sin.f64 47.4

       \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

4. Applied rewrites 47.4%

   \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

5. Taylor expanded in kx around 0

   \[\leadsto th \]
6. Step-by-step derivation

7. Applied rewrites 13.3%

   \[\leadsto th \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(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)))