Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.3% → 99.7%

Time: 6.5s

Alternatives: 22

Speedup: 1.2×

Specification

Math

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

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: 94.3% accurate, 1.0× speedup

Math

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

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

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

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 94.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

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: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\\ t_3 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_4 := \mathsf{fma}\left(\left(\left|ky\right| \cdot \left|ky\right|\right) \cdot \left|ky\right|, -0.16666666666666666, \left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.99998:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 0.01:\\ \;\;\;\;\frac{t\_4}{\mathsf{hypot}\left(t\_4, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9925:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0)))))
        (t_3 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_4
         (fma
          (* (* (fabs ky) (fabs ky)) (fabs ky))
          -0.16666666666666666
          (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.99998)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_3 -0.1)
        (* (/ t_2 (hypot (sin kx) t_1)) t_1)
        (if (<= t_3 0.01)
          (* (/ t_4 (hypot t_4 (sin kx))) (sin th))
          (if (<= t_3 0.9925)
            (* (/ t_1 (hypot t_1 (sin kx))) t_2)
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = th * (1.0 + (-0.16666666666666666 * pow(th, 2.0)));
	double t_3 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_4 = fma(((fabs(ky) * fabs(ky)) * fabs(ky)), -0.16666666666666666, fabs(ky));
	double tmp;
	if (t_3 <= -0.99998) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (t_2 / hypot(sin(kx), t_1)) * t_1;
	} else if (t_3 <= 0.01) {
		tmp = (t_4 / hypot(t_4, sin(kx))) * sin(th);
	} else if (t_3 <= 0.9925) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * t_2;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0))))
	t_3 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_4 = fma(Float64(Float64(abs(ky) * abs(ky)) * abs(ky)), -0.16666666666666666, abs(ky))
	tmp = 0.0
	if (t_3 <= -0.99998)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(t_2 / hypot(sin(kx), t_1)) * t_1);
	elseif (t_3 <= 0.01)
		tmp = Float64(Float64(t_4 / hypot(t_4, sin(kx))) * sin(th));
	elseif (t_3 <= 0.9925)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * t_2);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[Abs[ky], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.99998], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], N[(N[(t$95$2 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.01], N[(N[(t$95$4 / N[Sqrt[t$95$4 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9925], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $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))))) < -0.99997999999999998

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   if -0.99997999999999998 < (/.f64 (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 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.8

         \[\leadsto \frac{th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{\color{blue}{2}}\right)}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

   6. Applied rewrites 51.8%

      \[\leadsto \frac{\color{blue}{th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)}}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]

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

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.7

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   6. Applied rewrites 50.7%

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.7

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right), \sin kx\right)} \cdot \sin th \]

   9. Applied rewrites 54.7%

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. cube-unmult N/A

           \[\leadsto \frac{\mathsf{fma}\left({ky}^{3}, \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

       13. metadata-eval N/A

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

       14. pow-plus N/A

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

       15. lift-pow.f64 N/A

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

       16. lower-*.f64 54.6

           \[\leadsto \frac{\mathsf{fma}\left({ky}^{2} \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

       17. lift-pow.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 54.6

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

   11. Applied rewrites 54.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, \color{blue}{-0.16666666666666666}, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. cube-unmult N/A

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{3}, \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

       13. metadata-eval N/A

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

       14. pow-plus N/A

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

       15. lift-pow.f64 N/A

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

       16. lower-*.f64 54.7

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{2} \cdot ky, -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

       17. lift-pow.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 54.7

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

   13. Applied rewrites 54.7%

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

   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))))) < 0.992500000000000049

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.8

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{\color{blue}{2}}\right)\right) \]

   6. Applied rewrites 51.8%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]

   if 0.992500000000000049 < (/.f64 (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 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 3: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\\ t_3 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_4 := \mathsf{fma}\left(\left(\left|ky\right| \cdot \left|ky\right|\right) \cdot \left|ky\right|, -0.16666666666666666, \left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.99998:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.01:\\ \;\;\;\;\frac{t\_4}{\mathsf{hypot}\left(t\_4, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9925:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2
         (*
          (/ t_1 (hypot t_1 (sin kx)))
          (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0))))))
        (t_3 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_4
         (fma
          (* (* (fabs ky) (fabs ky)) (fabs ky))
          -0.16666666666666666
          (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.99998)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_3 -0.1)
        t_2
        (if (<= t_3 0.01)
          (* (/ t_4 (hypot t_4 (sin kx))) (sin th))
          (if (<= t_3 0.9925)
            t_2
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = (t_1 / hypot(t_1, sin(kx))) * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0))));
	double t_3 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_4 = fma(((fabs(ky) * fabs(ky)) * fabs(ky)), -0.16666666666666666, fabs(ky));
	double tmp;
	if (t_3 <= -0.99998) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = t_2;
	} else if (t_3 <= 0.01) {
		tmp = (t_4 / hypot(t_4, sin(kx))) * sin(th);
	} else if (t_3 <= 0.9925) {
		tmp = t_2;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))))
	t_3 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_4 = fma(Float64(Float64(abs(ky) * abs(ky)) * abs(ky)), -0.16666666666666666, abs(ky))
	tmp = 0.0
	if (t_3 <= -0.99998)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = t_2;
	elseif (t_3 <= 0.01)
		tmp = Float64(Float64(t_4 / hypot(t_4, sin(kx))) * sin(th));
	elseif (t_3 <= 0.9925)
		tmp = t_2;
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[Abs[ky], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.99998], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], t$95$2, If[LessEqual[t$95$3, 0.01], N[(N[(t$95$4 / N[Sqrt[t$95$4 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9925], t$95$2, N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99997999999999998

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   if -0.99997999999999998 < (/.f64 (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))))) < 0.992500000000000049

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 51.8

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{\color{blue}{2}}\right)\right) \]

   6. Applied rewrites 51.8%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]

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

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.7

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   6. Applied rewrites 50.7%

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.7

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right), \sin kx\right)} \cdot \sin th \]

   9. Applied rewrites 54.7%

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. cube-unmult N/A

           \[\leadsto \frac{\mathsf{fma}\left({ky}^{3}, \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

       13. metadata-eval N/A

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

       14. pow-plus N/A

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

       15. lift-pow.f64 N/A

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

       16. lower-*.f64 54.6

           \[\leadsto \frac{\mathsf{fma}\left({ky}^{2} \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

       17. lift-pow.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 54.6

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

   11. Applied rewrites 54.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, \color{blue}{-0.16666666666666666}, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. cube-unmult N/A

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{3}, \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

       13. metadata-eval N/A

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

       14. pow-plus N/A

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

       15. lift-pow.f64 N/A

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

       16. lower-*.f64 54.7

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{2} \cdot ky, -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

       17. lift-pow.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 54.7

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

   13. Applied rewrites 54.7%

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

   if 0.992500000000000049 < (/.f64 (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 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 4: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\left(\left|ky\right| \cdot \left|ky\right|\right) \cdot \left|ky\right|, -0.16666666666666666, \left|ky\right|\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \frac{t\_2}{\sqrt{{\sin kx}^{2} + {t\_2}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.99998:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(t\_2, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, t\_2\right)} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq 0.01:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9925:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(t\_2, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (fma
          (* (* (fabs ky) (fabs ky)) (fabs ky))
          -0.16666666666666666
          (fabs ky)))
        (t_2 (sin (fabs ky)))
        (t_3 (/ t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_2 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.99998)
      (* (/ t_2 (hypot t_2 kx)) (sin th))
      (if (<= t_3 -0.1)
        (* (/ th (hypot (sin kx) t_2)) t_2)
        (if (<= t_3 0.01)
          (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
          (if (<= t_3 0.9925)
            (* (/ t_2 (hypot t_2 (sin kx))) th)
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma(((fabs(ky) * fabs(ky)) * fabs(ky)), -0.16666666666666666, fabs(ky));
	double t_2 = sin(fabs(ky));
	double t_3 = t_2 / sqrt((pow(sin(kx), 2.0) + pow(t_2, 2.0)));
	double tmp;
	if (t_3 <= -0.99998) {
		tmp = (t_2 / hypot(t_2, kx)) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (th / hypot(sin(kx), t_2)) * t_2;
	} else if (t_3 <= 0.01) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else if (t_3 <= 0.9925) {
		tmp = (t_2 / hypot(t_2, sin(kx))) * th;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(Float64(Float64(abs(ky) * abs(ky)) * abs(ky)), -0.16666666666666666, abs(ky))
	t_2 = sin(abs(ky))
	t_3 = Float64(t_2 / sqrt(Float64((sin(kx) ^ 2.0) + (t_2 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.99998)
		tmp = Float64(Float64(t_2 / hypot(t_2, kx)) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(th / hypot(sin(kx), t_2)) * t_2);
	elseif (t_3 <= 0.01)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	elseif (t_3 <= 0.9925)
		tmp = Float64(Float64(t_2 / hypot(t_2, sin(kx))) * th);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[Abs[ky], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.99998], N[(N[(t$95$2 / N[Sqrt[t$95$2 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 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$3, 0.9925], N[(N[(t$95$2 / N[Sqrt[t$95$2 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $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))))) < -0.99997999999999998

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   if -0.99997999999999998 < (/.f64 (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 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 52.0%

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

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

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.7

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   6. Applied rewrites 50.7%

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.7

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right), \sin kx\right)} \cdot \sin th \]

   9. Applied rewrites 54.7%

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. cube-unmult N/A

           \[\leadsto \frac{\mathsf{fma}\left({ky}^{3}, \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

       13. metadata-eval N/A

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

       14. pow-plus N/A

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

       15. lift-pow.f64 N/A

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

       16. lower-*.f64 54.6

           \[\leadsto \frac{\mathsf{fma}\left({ky}^{2} \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

       17. lift-pow.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 54.6

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

   11. Applied rewrites 54.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, \color{blue}{-0.16666666666666666}, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. cube-unmult N/A

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{3}, \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

       13. metadata-eval N/A

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

       14. pow-plus N/A

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

       15. lift-pow.f64 N/A

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

       16. lower-*.f64 54.7

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{2} \cdot ky, -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

       17. lift-pow.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 54.7

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

   13. Applied rewrites 54.7%

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

   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))))) < 0.992500000000000049

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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

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

   6. Applied rewrites 52.0%

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

   if 0.992500000000000049 < (/.f64 (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 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 5: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\left(\left|ky\right| \cdot \left|ky\right|\right) \cdot \left|ky\right|, -0.16666666666666666, \left|ky\right|\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \frac{t\_2}{\sqrt{{\sin kx}^{2} + {t\_2}^{2}}}\\ t_4 := \frac{th}{\mathsf{hypot}\left(\sin kx, t\_2\right)} \cdot t\_2\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.99998:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(t\_2, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.01:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9925:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (fma
          (* (* (fabs ky) (fabs ky)) (fabs ky))
          -0.16666666666666666
          (fabs ky)))
        (t_2 (sin (fabs ky)))
        (t_3 (/ t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_2 2.0)))))
        (t_4 (* (/ th (hypot (sin kx) t_2)) t_2)))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.99998)
      (* (/ t_2 (hypot t_2 kx)) (sin th))
      (if (<= t_3 -0.1)
        t_4
        (if (<= t_3 0.01)
          (* (/ t_1 (hypot t_1 (sin kx))) (sin th))
          (if (<= t_3 0.9925)
            t_4
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma(((fabs(ky) * fabs(ky)) * fabs(ky)), -0.16666666666666666, fabs(ky));
	double t_2 = sin(fabs(ky));
	double t_3 = t_2 / sqrt((pow(sin(kx), 2.0) + pow(t_2, 2.0)));
	double t_4 = (th / hypot(sin(kx), t_2)) * t_2;
	double tmp;
	if (t_3 <= -0.99998) {
		tmp = (t_2 / hypot(t_2, kx)) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = t_4;
	} else if (t_3 <= 0.01) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * sin(th);
	} else if (t_3 <= 0.9925) {
		tmp = t_4;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(Float64(Float64(abs(ky) * abs(ky)) * abs(ky)), -0.16666666666666666, abs(ky))
	t_2 = sin(abs(ky))
	t_3 = Float64(t_2 / sqrt(Float64((sin(kx) ^ 2.0) + (t_2 ^ 2.0))))
	t_4 = Float64(Float64(th / hypot(sin(kx), t_2)) * t_2)
	tmp = 0.0
	if (t_3 <= -0.99998)
		tmp = Float64(Float64(t_2 / hypot(t_2, kx)) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = t_4;
	elseif (t_3 <= 0.01)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * sin(th));
	elseif (t_3 <= 0.9925)
		tmp = t_4;
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[Abs[ky], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.99998], N[(N[(t$95$2 / N[Sqrt[t$95$2 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], t$95$4, If[LessEqual[t$95$3, 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$3, 0.9925], t$95$4, N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99997999999999998

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   if -0.99997999999999998 < (/.f64 (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))))) < 0.992500000000000049

   1. Initial program 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 52.0%

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

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

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.7

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   6. Applied rewrites 50.7%

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.7

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right), \sin kx\right)} \cdot \sin th \]

   9. Applied rewrites 54.7%

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. cube-unmult N/A

           \[\leadsto \frac{\mathsf{fma}\left({ky}^{3}, \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

       13. metadata-eval N/A

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

       14. pow-plus N/A

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

       15. lift-pow.f64 N/A

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

       16. lower-*.f64 54.6

           \[\leadsto \frac{\mathsf{fma}\left({ky}^{2} \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

       17. lift-pow.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 54.6

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

   11. Applied rewrites 54.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, \color{blue}{-0.16666666666666666}, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. cube-unmult N/A

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{3}, \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

       13. metadata-eval N/A

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

       14. pow-plus N/A

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

       15. lift-pow.f64 N/A

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

       16. lower-*.f64 54.7

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{2} \cdot ky, -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

       17. lift-pow.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 54.7

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

   13. Applied rewrites 54.7%

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

   if 0.992500000000000049 < (/.f64 (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 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 6: 86.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ t_3 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_4 := \mathsf{fma}\left(\left(\left|ky\right| \cdot \left|ky\right|\right) \cdot \left|ky\right|, -0.16666666666666666, \left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.99998:\\ \;\;\;\;\sin th \cdot \frac{t\_1}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(\left|ky\right| + \left|ky\right|\right), 0.5\right)}}\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.01:\\ \;\;\;\;\frac{t\_4}{\mathsf{hypot}\left(t\_4, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9925:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (* (/ th (hypot (sin kx) t_1)) t_1))
        (t_3 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_4
         (fma
          (* (* (fabs ky) (fabs ky)) (fabs ky))
          -0.16666666666666666
          (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.99998)
      (* (sin th) (/ t_1 (sqrt (fma -0.5 (cos (+ (fabs ky) (fabs ky))) 0.5))))
      (if (<= t_3 -0.1)
        t_2
        (if (<= t_3 0.01)
          (* (/ t_4 (hypot t_4 (sin kx))) (sin th))
          (if (<= t_3 0.9925)
            t_2
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = (th / hypot(sin(kx), t_1)) * t_1;
	double t_3 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_4 = fma(((fabs(ky) * fabs(ky)) * fabs(ky)), -0.16666666666666666, fabs(ky));
	double tmp;
	if (t_3 <= -0.99998) {
		tmp = sin(th) * (t_1 / sqrt(fma(-0.5, cos((fabs(ky) + fabs(ky))), 0.5)));
	} else if (t_3 <= -0.1) {
		tmp = t_2;
	} else if (t_3 <= 0.01) {
		tmp = (t_4 / hypot(t_4, sin(kx))) * sin(th);
	} else if (t_3 <= 0.9925) {
		tmp = t_2;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(Float64(th / hypot(sin(kx), t_1)) * t_1)
	t_3 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_4 = fma(Float64(Float64(abs(ky) * abs(ky)) * abs(ky)), -0.16666666666666666, abs(ky))
	tmp = 0.0
	if (t_3 <= -0.99998)
		tmp = Float64(sin(th) * Float64(t_1 / sqrt(fma(-0.5, cos(Float64(abs(ky) + abs(ky))), 0.5))));
	elseif (t_3 <= -0.1)
		tmp = t_2;
	elseif (t_3 <= 0.01)
		tmp = Float64(Float64(t_4 / hypot(t_4, sin(kx))) * sin(th));
	elseif (t_3 <= 0.9925)
		tmp = t_2;
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[Abs[ky], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.99998], N[(N[Sin[th], $MachinePrecision] * N[(t$95$1 / N[Sqrt[N[(-0.5 * N[Cos[N[(N[Abs[ky], $MachinePrecision] + N[Abs[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], t$95$2, If[LessEqual[t$95$3, 0.01], N[(N[(t$95$4 / N[Sqrt[t$95$4 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9925], t$95$2, N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99997999999999998

   1. Initial program 94.3%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6

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

   4. Applied rewrites 40.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   6. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}{\sin th \cdot \sin ky}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 31.0

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

   8. Applied rewrites 31.0%

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

   if -0.99997999999999998 < (/.f64 (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))))) < 0.992500000000000049

   1. Initial program 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 52.0%

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

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

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.7

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   6. Applied rewrites 50.7%

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 54.7

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right), \sin kx\right)} \cdot \sin th \]

   9. Applied rewrites 54.7%

      \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. cube-unmult N/A

           \[\leadsto \frac{\mathsf{fma}\left({ky}^{3}, \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

       13. metadata-eval N/A

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

       14. pow-plus N/A

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

       15. lift-pow.f64 N/A

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

       16. lower-*.f64 54.6

           \[\leadsto \frac{\mathsf{fma}\left({ky}^{2} \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

       17. lift-pow.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 54.6

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]

   11. Applied rewrites 54.6%

       \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, \color{blue}{-0.16666666666666666}, ky\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), \sin kx\right)} \cdot \sin th \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. lift-pow.f64 N/A

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

       11. unpow2 N/A

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

       12. cube-unmult N/A

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{3}, \frac{-1}{6}, ky\right), \sin kx\right)} \cdot \sin th \]

       13. metadata-eval N/A

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

       14. pow-plus N/A

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

       15. lift-pow.f64 N/A

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

       16. lower-*.f64 54.7

           \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot ky, -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(\mathsf{fma}\left({ky}^{2} \cdot ky, -0.16666666666666666, ky\right), \sin kx\right)} \cdot \sin th \]

       17. lift-pow.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 54.7

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

   13. Applied rewrites 54.7%

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

   if 0.992500000000000049 < (/.f64 (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 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 7: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq -0.1:\\ \;\;\;\;\sin th \cdot \frac{t\_1}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(\left|ky\right| + \left|ky\right|\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) -0.1)
      (* (sin th) (/ t_1 (sqrt (fma -0.5 (cos (+ (fabs ky) (fabs ky))) 0.5))))
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

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

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= -0.1)
		tmp = Float64(sin(th) * Float64(t_1 / sqrt(fma(-0.5, cos(Float64(abs(ky) + abs(ky))), 0.5))));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.1], N[(N[Sin[th], $MachinePrecision] * N[(t$95$1 / N[Sqrt[N[(-0.5 * N[Cos[N[(N[Abs[ky], $MachinePrecision] + N[Abs[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.3%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6

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

   4. Applied rewrites 40.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   6. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}{\sin th \cdot \sin ky}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 31.0

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

   8. Applied rewrites 31.0%

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

   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)))))

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 8: 78.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq -0.1:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(-0.5, \cos \left(\left|ky\right| + \left|ky\right|\right), 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) -0.1)
      (* t_1 (/ (sin th) (sqrt (fma -0.5 (cos (+ (fabs ky) (fabs ky))) 0.5))))
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

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

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= -0.1)
		tmp = Float64(t_1 * Float64(sin(th) / sqrt(fma(-0.5, cos(Float64(abs(ky) + abs(ky))), 0.5))));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.1], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(-0.5 * N[Cos[N[(N[Abs[ky], $MachinePrecision] + N[Abs[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.3%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6

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

   4. Applied rewrites 40.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   6. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}{\sin th \cdot \sin ky}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

   8. Applied rewrites 31.0%

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

   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)))))

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 9: 71.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\frac{t\_1}{\sqrt{{t\_1}^{2}}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_1 -0.05)
      (* (/ t_1 (sqrt (pow t_1 2.0))) th)
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (t_1 / sqrt(pow(t_1, 2.0))) * th;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (t_1 / Math.sqrt(Math.pow(t_1, 2.0))) * th;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if t_1 <= -0.05:
		tmp = (t_1 / math.sqrt(math.pow(t_1, 2.0))) * th
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(Float64(t_1 / sqrt((t_1 ^ 2.0))) * th);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if (t_1 <= -0.05)
		tmp = (t_1 / sqrt((t_1 ^ 2.0))) * th;
	else
		tmp = (abs(ky) / hypot(abs(ky), sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -0.05], N[(N[(t$95$1 / N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.050000000000000003

   1. Initial program 94.3%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6

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

   4. Applied rewrites 40.6%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 22.2%

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

   if -0.050000000000000003 < (sin.f64 ky)

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 10: 71.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\frac{1}{\frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\left|ky\right| + \left|ky\right|\right)}}{th \cdot t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_1 -0.05)
      (/
       1.0
       (/ (sqrt (- 0.5 (* 0.5 (cos (+ (fabs ky) (fabs ky)))))) (* th t_1)))
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = 1.0 / (sqrt((0.5 - (0.5 * cos((fabs(ky) + fabs(ky)))))) / (th * t_1));
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = 1.0 / (Math.sqrt((0.5 - (0.5 * Math.cos((Math.abs(ky) + Math.abs(ky)))))) / (th * t_1));
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if t_1 <= -0.05:
		tmp = 1.0 / (math.sqrt((0.5 - (0.5 * math.cos((math.fabs(ky) + math.fabs(ky)))))) / (th * t_1))
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(1.0 / Float64(sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(abs(ky) + abs(ky)))))) / Float64(th * t_1)));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if (t_1 <= -0.05)
		tmp = 1.0 / (sqrt((0.5 - (0.5 * cos((abs(ky) + abs(ky)))))) / (th * t_1));
	else
		tmp = (abs(ky) / hypot(abs(ky), sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -0.05], N[(1.0 / N[(N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(N[Abs[ky], $MachinePrecision] + N[Abs[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(th * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.050000000000000003

   1. Initial program 94.3%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6

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

   4. Applied rewrites 40.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   6. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}{\sin th \cdot \sin ky}}} \]

   7. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 17.0

         \[\leadsto \frac{1}{\frac{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}{th \cdot \sin ky}} \]

   9. Applied rewrites 17.0%

      \[\leadsto \frac{1}{\frac{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}{\color{blue}{th \cdot \sin ky}}} \]

   if -0.050000000000000003 < (sin.f64 ky)

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 11: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_1 -0.05)
      (* (/ t_1 (hypot t_1 kx)) th)
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (t_1 / hypot(t_1, kx)) * th;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * th;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if t_1 <= -0.05:
		tmp = (t_1 / math.hypot(t_1, kx)) * th
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * th);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if (t_1 <= -0.05)
		tmp = (t_1 / hypot(t_1, kx)) * th;
	else
		tmp = (abs(ky) / hypot(abs(ky), sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -0.05], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.050000000000000003

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 34.3%

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

   if -0.050000000000000003 < (sin.f64 ky)

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 12: 68.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left|ky\right|, \left|ky\right|, 1\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_2}{\sqrt{{\sin kx}^{2} + {t\_2}^{2}}} \leq -0.15:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (* (fabs ky) (fma (* -0.16666666666666666 (fabs ky)) (fabs ky) 1.0)))
        (t_2 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_2 2.0)))) -0.15)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * fma((-0.16666666666666666 * fabs(ky)), fabs(ky), 1.0);
	double t_2 = sin(fabs(ky));
	double tmp;
	if ((t_2 / sqrt((pow(sin(kx), 2.0) + pow(t_2, 2.0)))) <= -0.15) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(abs(ky) * fma(Float64(-0.16666666666666666 * abs(ky)), abs(ky), 1.0))
	t_2 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_2 / sqrt(Float64((sin(kx) ^ 2.0) + (t_2 ^ 2.0)))) <= -0.15)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Abs[ky], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$2 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.15], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.149999999999999994

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 32.8

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right)}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   9. Applied rewrites 32.8%

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 36.4

          \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 36.4%

       \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   13. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 36.3

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot ky, ky, 1\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), kx\right)} \cdot \sin th \]

   14. Applied rewrites 36.3%

       \[\leadsto \frac{ky \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot ky, \color{blue}{ky}, 1\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), kx\right)} \cdot \sin th \]
   15. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 36.4

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

   16. Applied rewrites 36.4%

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

   if -0.149999999999999994 < (/.f64 (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 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 50.9%

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

   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 64.3%

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

Alternative 13: 67.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \left|ky\right| \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left|ky\right|, \left|ky\right|, 1\right)\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \frac{t\_2}{\sqrt{{\sin kx}^{2} + {t\_2}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.15:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (* (fabs ky) (fma (* -0.16666666666666666 (fabs ky)) (fabs ky) 1.0)))
        (t_2 (sin (fabs ky)))
        (t_3 (/ t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_2 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.15)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_3 0.01)
        (* (sin th) (/ (fabs ky) (fabs (sin kx))))
        (* (/ (fabs ky) (hypot (fabs ky) kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fabs(ky) * fma((-0.16666666666666666 * fabs(ky)), fabs(ky), 1.0);
	double t_2 = sin(fabs(ky));
	double t_3 = t_2 / sqrt((pow(sin(kx), 2.0) + pow(t_2, 2.0)));
	double tmp;
	if (t_3 <= -0.15) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_3 <= 0.01) {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(abs(ky) * fma(Float64(-0.16666666666666666 * abs(ky)), abs(ky), 1.0))
	t_2 = sin(abs(ky))
	t_3 = Float64(t_2 / sqrt(Float64((sin(kx) ^ 2.0) + (t_2 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.15)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_3 <= 0.01)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Abs[ky], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.15], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.01], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.149999999999999994

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 32.8

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right)}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   9. Applied rewrites 32.8%

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 36.4

          \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 36.4%

       \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   13. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 36.3

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot ky, ky, 1\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), kx\right)} \cdot \sin th \]

   14. Applied rewrites 36.3%

       \[\leadsto \frac{ky \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot ky, \color{blue}{ky}, 1\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right), kx\right)} \cdot \sin th \]
   15. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 36.4

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

   16. Applied rewrites 36.4%

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

   if -0.149999999999999994 < (/.f64 (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 94.3%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 35.5

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

   4. Applied rewrites 35.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 36.4

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

      9. lift-sqrt.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

          \[\leadsto \sin th \cdot \frac{ky}{\left|\sin kx\right|} \]

      13. lower-fabs.f64 39.1

          \[\leadsto \sin th \cdot \frac{ky}{\left|\sin kx\right|} \]

   6. Applied rewrites 39.1%

      \[\leadsto \sin th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

   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 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 33.3%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 45.6%

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

Alternative 14: 67.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\left|ky\right| \cdot \left|ky\right|, -0.16666666666666666, 1\right) \cdot \left|ky\right|\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \frac{t\_2}{\sqrt{{\sin kx}^{2} + {t\_2}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.15:\\ \;\;\;\;\frac{t\_1 \cdot \sin th}{\mathsf{hypot}\left(kx, t\_1\right)}\\ \mathbf{elif}\;t\_3 \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (* (fma (* (fabs ky) (fabs ky)) -0.16666666666666666 1.0) (fabs ky)))
        (t_2 (sin (fabs ky)))
        (t_3 (/ t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_2 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.15)
      (/ (* t_1 (sin th)) (hypot kx t_1))
      (if (<= t_3 0.01)
        (* (sin th) (/ (fabs ky) (fabs (sin kx))))
        (* (/ (fabs ky) (hypot (fabs ky) kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((fabs(ky) * fabs(ky)), -0.16666666666666666, 1.0) * fabs(ky);
	double t_2 = sin(fabs(ky));
	double t_3 = t_2 / sqrt((pow(sin(kx), 2.0) + pow(t_2, 2.0)));
	double tmp;
	if (t_3 <= -0.15) {
		tmp = (t_1 * sin(th)) / hypot(kx, t_1);
	} else if (t_3 <= 0.01) {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(abs(ky) * abs(ky)), -0.16666666666666666, 1.0) * abs(ky))
	t_2 = sin(abs(ky))
	t_3 = Float64(t_2 / sqrt(Float64((sin(kx) ^ 2.0) + (t_2 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.15)
		tmp = Float64(Float64(t_1 * sin(th)) / hypot(kx, t_1));
	elseif (t_3 <= 0.01)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.15], N[(N[(t$95$1 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[kx ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.01], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.149999999999999994

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 32.8

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right)}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   9. Applied rewrites 32.8%

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 36.4

          \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 36.4%

       \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   13. Step-by-step derivation

       1. lift-sin.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. lower-/.f64 N/A

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

   14. Applied rewrites 33.1%

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

   if -0.149999999999999994 < (/.f64 (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 94.3%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 35.5

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

   4. Applied rewrites 35.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 36.4

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

      9. lift-sqrt.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

          \[\leadsto \sin th \cdot \frac{ky}{\left|\sin kx\right|} \]

      13. lower-fabs.f64 39.1

          \[\leadsto \sin th \cdot \frac{ky}{\left|\sin kx\right|} \]

   6. Applied rewrites 39.1%

      \[\leadsto \sin th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

   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 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 33.3%

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

   10. Taylor expanded in ky around 0

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

   12. Applied rewrites 45.6%

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

Alternative 15: 66.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\left|ky\right| \cdot \left|ky\right|, -0.16666666666666666, 1\right) \cdot \left|ky\right|\\ t_2 := \sin \left(\left|ky\right|\right)\\ t_3 := \frac{t\_2}{\sqrt{{\sin kx}^{2} + {t\_2}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.15:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\mathsf{hypot}\left(kx, t\_1\right)}\\ \mathbf{elif}\;t\_3 \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (* (fma (* (fabs ky) (fabs ky)) -0.16666666666666666 1.0) (fabs ky)))
        (t_2 (sin (fabs ky)))
        (t_3 (/ t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_2 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -0.15)
      (* t_1 (/ (sin th) (hypot kx t_1)))
      (if (<= t_3 0.01)
        (* (sin th) (/ (fabs ky) (fabs (sin kx))))
        (* (/ (fabs ky) (hypot (fabs ky) kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((fabs(ky) * fabs(ky)), -0.16666666666666666, 1.0) * fabs(ky);
	double t_2 = sin(fabs(ky));
	double t_3 = t_2 / sqrt((pow(sin(kx), 2.0) + pow(t_2, 2.0)));
	double tmp;
	if (t_3 <= -0.15) {
		tmp = t_1 * (sin(th) / hypot(kx, t_1));
	} else if (t_3 <= 0.01) {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(abs(ky) * abs(ky)), -0.16666666666666666, 1.0) * abs(ky))
	t_2 = sin(abs(ky))
	t_3 = Float64(t_2 / sqrt(Float64((sin(kx) ^ 2.0) + (t_2 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.15)
		tmp = Float64(t_1 * Float64(sin(th) / hypot(kx, t_1)));
	elseif (t_3 <= 0.01)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(N[Abs[ky], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -0.15], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[kx ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.01], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.149999999999999994

   1. Initial program 94.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   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 57.5%

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

   7. Taylor expanded in ky around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 32.8

         \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right)}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   9. Applied rewrites 32.8%

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 36.4

          \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{\color{blue}{2}}\right), kx\right)} \cdot \sin th \]

   12. Applied rewrites 36.4%

       \[\leadsto \frac{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + -0.16666666666666666 \cdot {ky}^{2}\right)}, kx\right)} \cdot \sin th \]
   13. Step-by-step derivation

       1. lift-sin.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

   14. Applied rewrites 35.4%

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

   if -0.149999999999999994 < (/.f64 (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 94.3%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 35.5

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

   4. Applied rewrites 35.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 36.4

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

      9. lift-sqrt.f64 N/A

         \[\leadsto \sin th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \sin th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \sin th \cdot \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \sin th \cdot \frac{ky}{\left|\sin kx\right|} \]

      13. lower-fabs.f64 39.1

          \[\leadsto \sin th \cdot \frac{ky}{\left|\sin kx\right|} \]

   6. Applied rewrites 39.1%

      \[\leadsto \sin th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

   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 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   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 57.5%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 33.3%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

   12. Applied rewrites 45.6%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 0.01)
      (* (sin th) (/ (fabs ky) (fabs (sin kx))))
      (* (/ (fabs ky) (hypot (fabs ky) kx)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 0.01) {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 0.01) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(Math.sin(kx)));
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), kx)) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 0.01:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(math.sin(kx)))
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), kx)) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.01)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.01)
		tmp = sin(th) * (abs(ky) / abs(sin(kx)));
	else
		tmp = (abs(ky) / hypot(abs(ky), kx)) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.5

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. mult-flip N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \color{blue}{\frac{1}{\sqrt{{\sin kx}^{2}}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(ky \cdot \sin th\right) \cdot \frac{\color{blue}{1}}{\sqrt{{\sin kx}^{2}}} \]

      4. *-commutative N/A

         \[\leadsto \left(\sin th \cdot ky\right) \cdot \frac{\color{blue}{1}}{\sqrt{{\sin kx}^{2}}} \]

      5. associate-*l* N/A

         \[\leadsto \sin th \cdot \color{blue}{\left(ky \cdot \frac{1}{\sqrt{{\sin kx}^{2}}}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\left(ky \cdot \frac{1}{\sqrt{{\sin kx}^{2}}}\right)} \]

      7. mult-flip-rev N/A

         \[\leadsto \sin th \cdot \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      8. lower-/.f64 36.4

         \[\leadsto \sin th \cdot \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \sin th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \sin th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}} \]

      11. pow2 N/A

          \[\leadsto \sin th \cdot \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      12. rem-sqrt-square-rev N/A

          \[\leadsto \sin th \cdot \frac{ky}{\left|\sin kx\right|} \]

      13. lower-fabs.f64 39.1

          \[\leadsto \sin th \cdot \frac{ky}{\left|\sin kx\right|} \]

   6. Applied rewrites 39.1%

      \[\leadsto \sin th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

   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 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   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 57.5%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 33.3%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

   12. Applied rewrites 45.6%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{\sin th}{\left|\sin kx\right|} \cdot \left|ky\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 0.01)
      (* (/ (sin th) (fabs (sin kx))) (fabs ky))
      (* (/ (fabs ky) (hypot (fabs ky) kx)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 0.01) {
		tmp = (sin(th) / fabs(sin(kx))) * fabs(ky);
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 0.01) {
		tmp = (Math.sin(th) / Math.abs(Math.sin(kx))) * Math.abs(ky);
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), kx)) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 0.01:
		tmp = (math.sin(th) / math.fabs(math.sin(kx))) * math.fabs(ky)
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), kx)) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.01)
		tmp = Float64(Float64(sin(th) / abs(sin(kx))) * abs(ky));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 0.01)
		tmp = (sin(th) / abs(sin(kx))) * abs(ky);
	else
		tmp = (abs(ky) / hypot(abs(ky), kx)) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.5

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. associate-/l* N/A

         \[\leadsto ky \cdot \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2}}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin th}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{ky} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{ky} \]

      6. lower-/.f64 36.4

         \[\leadsto \frac{\sin th}{\sqrt{{\sin kx}^{2}}} \cdot ky \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{{\sin kx}^{2}}} \cdot ky \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{{\sin kx}^{2}}} \cdot ky \]

      9. pow2 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\sin kx \cdot \sin kx}} \cdot ky \]

      10. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\sin th}{\left|\sin kx\right|} \cdot ky \]

      11. lower-fabs.f64 39.1

          \[\leadsto \frac{\sin th}{\left|\sin kx\right|} \cdot ky \]

   6. Applied rewrites 39.1%

      \[\leadsto \frac{\sin th}{\left|\sin kx\right|} \cdot \color{blue}{ky} \]

   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 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   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 57.5%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 33.3%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

   12. Applied rewrites 45.6%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 49.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.09:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{th}{\left|\sin kx\right|}\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 0.09)
   (* (/ ky (hypot ky kx)) (sin th))
   (* ky (/ th (fabs (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 0.09) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = ky * (th / fabs(sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(kx), 2.0) <= 0.09) {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	} else {
		tmp = ky * (th / Math.abs(Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(kx), 2.0) <= 0.09:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	else:
		tmp = ky * (th / math.fabs(math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 0.09)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(ky * Float64(th / abs(sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(kx) ^ 2.0) <= 0.09)
		tmp = (ky / hypot(ky, kx)) * sin(th);
	else
		tmp = ky * (th / abs(sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 0.09], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(ky * N[(th / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.089999999999999997

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   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 57.5%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 33.3%

      \[\leadsto \frac{\color{blue}{ky}}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th \]

   10. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]
   11. Step-by-step derivation

   12. Applied rewrites 45.6%

       \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, kx\right)} \cdot \sin th \]

   if 0.089999999999999997 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.5

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 18.8%

      \[\leadsto \frac{ky \cdot th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{ky \cdot th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{ky \cdot th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. associate-/l* N/A

         \[\leadsto ky \cdot \color{blue}{\frac{th}{\sqrt{{\sin kx}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto ky \cdot \color{blue}{\frac{th}{\sqrt{{\sin kx}^{2}}}} \]

      5. lower-/.f64 19.7

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      8. pow2 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{\sin kx \cdot \sin kx}} \]

      9. rem-sqrt-square-rev N/A

         \[\leadsto ky \cdot \frac{th}{\left|\sin kx\right|} \]

      10. lower-fabs.f64 21.1

          \[\leadsto ky \cdot \frac{th}{\left|\sin kx\right|} \]

   9. Applied rewrites 21.1%

      \[\leadsto ky \cdot \color{blue}{\frac{th}{\left|\sin kx\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 26.7% accurate, 2.7× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 1.35 \cdot 10^{-24}:\\ \;\;\;\;ky \cdot \frac{\left|th\right|}{\left|\sin \left(\left|kx\right|\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin \left(\left|th\right|\right)}{\frac{\left|kx\right| \cdot \sqrt{2}}{\sqrt{2}}}\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (copysign 1.0 th)
  (if (<= (fabs th) 1.35e-24)
    (* ky (/ (fabs th) (fabs (sin (fabs kx)))))
    (/ (* ky (sin (fabs th))) (/ (* (fabs kx) (sqrt 2.0)) (sqrt 2.0))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 1.35e-24) {
		tmp = ky * (fabs(th) / fabs(sin(fabs(kx))));
	} else {
		tmp = (ky * sin(fabs(th))) / ((fabs(kx) * sqrt(2.0)) / sqrt(2.0));
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 1.35e-24) {
		tmp = ky * (Math.abs(th) / Math.abs(Math.sin(Math.abs(kx))));
	} else {
		tmp = (ky * Math.sin(Math.abs(th))) / ((Math.abs(kx) * Math.sqrt(2.0)) / Math.sqrt(2.0));
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 1.35e-24:
		tmp = ky * (math.fabs(th) / math.fabs(math.sin(math.fabs(kx))))
	else:
		tmp = (ky * math.sin(math.fabs(th))) / ((math.fabs(kx) * math.sqrt(2.0)) / math.sqrt(2.0))
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 1.35e-24)
		tmp = Float64(ky * Float64(abs(th) / abs(sin(abs(kx)))));
	else
		tmp = Float64(Float64(ky * sin(abs(th))) / Float64(Float64(abs(kx) * sqrt(2.0)) / sqrt(2.0)));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 1.35e-24)
		tmp = ky * (abs(th) / abs(sin(abs(kx))));
	else
		tmp = (ky * sin(abs(th))) / ((abs(kx) * sqrt(2.0)) / sqrt(2.0));
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 1.35e-24], N[(ky * N[(N[Abs[th], $MachinePrecision] / N[Abs[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky * N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Abs[kx], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 1.35000000000000003e-24

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.5

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 18.8%

      \[\leadsto \frac{ky \cdot th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{ky \cdot th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{ky \cdot th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. associate-/l* N/A

         \[\leadsto ky \cdot \color{blue}{\frac{th}{\sqrt{{\sin kx}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto ky \cdot \color{blue}{\frac{th}{\sqrt{{\sin kx}^{2}}}} \]

      5. lower-/.f64 19.7

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      8. pow2 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{\sin kx \cdot \sin kx}} \]

      9. rem-sqrt-square-rev N/A

         \[\leadsto ky \cdot \frac{th}{\left|\sin kx\right|} \]

      10. lower-fabs.f64 21.1

          \[\leadsto ky \cdot \frac{th}{\left|\sin kx\right|} \]

   9. Applied rewrites 21.1%

      \[\leadsto ky \cdot \color{blue}{\frac{th}{\left|\sin kx\right|}} \]

   if 1.35000000000000003e-24 < th

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.5

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      3. pow2 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\sin kx \cdot \sin kx}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\sin kx \cdot \sin kx}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\sin kx \cdot \sin kx}} \]

      6. sin-mult N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}} \]

      7. sqrt-div N/A

         \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}}{\color{blue}{\sqrt{2}}}} \]

      8. lower-unsound-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}}{\color{blue}{\sqrt{2}}}} \]

      9. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}}{\sqrt{\color{blue}{2}}}} \]

      10. sub-flip N/A

          \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{\cos \left(kx - kx\right) + \left(\mathsf{neg}\left(\cos \left(kx + kx\right)\right)\right)}}{\sqrt{2}}} \]

      11. +-inverses N/A

          \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{\cos 0 + \left(\mathsf{neg}\left(\cos \left(kx + kx\right)\right)\right)}}{\sqrt{2}}} \]

      12. cos-0 N/A

          \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{1 + \left(\mathsf{neg}\left(\cos \left(kx + kx\right)\right)\right)}}{\sqrt{2}}} \]

      13. sub-flip-reverse N/A

          \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{1 - \cos \left(kx + kx\right)}}{\sqrt{2}}} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{1 - \cos \left(kx + kx\right)}}{\sqrt{2}}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{1 - \cos \left(kx + kx\right)}}{\sqrt{2}}} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{1 - \cos \left(kx + kx\right)}}{\sqrt{2}}} \]

      17. lower-unsound-sqrt.f64 27.5

          \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{1 - \cos \left(kx + kx\right)}}{\sqrt{2}}} \]

   6. Applied rewrites 27.5%

      \[\leadsto \frac{ky \cdot \sin th}{\frac{\sqrt{1 - \cos \left(kx + kx\right)}}{\color{blue}{\sqrt{2}}}} \]

   7. Taylor expanded in kx around 0

      \[\leadsto \frac{ky \cdot \sin th}{\frac{kx \cdot \sqrt{2}}{\sqrt{\color{blue}{2}}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\frac{kx \cdot \sqrt{2}}{\sqrt{2}}} \]

      2. lower-sqrt.f64 16.0

         \[\leadsto \frac{ky \cdot \sin th}{\frac{kx \cdot \sqrt{2}}{\sqrt{2}}} \]

   9. Applied rewrites 16.0%

      \[\leadsto \frac{ky \cdot \sin th}{\frac{kx \cdot \sqrt{2}}{\sqrt{\color{blue}{2}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 26.7% accurate, 3.2× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 1.35 \cdot 10^{-24}:\\ \;\;\;\;ky \cdot \frac{\left|th\right|}{\left|\sin \left(\left|kx\right|\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin \left(\left|th\right|\right)}{\left|kx\right|}\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (copysign 1.0 th)
  (if (<= (fabs th) 1.35e-24)
    (* ky (/ (fabs th) (fabs (sin (fabs kx)))))
    (/ (* ky (sin (fabs th))) (fabs kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 1.35e-24) {
		tmp = ky * (fabs(th) / fabs(sin(fabs(kx))));
	} else {
		tmp = (ky * sin(fabs(th))) / fabs(kx);
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 1.35e-24) {
		tmp = ky * (Math.abs(th) / Math.abs(Math.sin(Math.abs(kx))));
	} else {
		tmp = (ky * Math.sin(Math.abs(th))) / Math.abs(kx);
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 1.35e-24:
		tmp = ky * (math.fabs(th) / math.fabs(math.sin(math.fabs(kx))))
	else:
		tmp = (ky * math.sin(math.fabs(th))) / math.fabs(kx)
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 1.35e-24)
		tmp = Float64(ky * Float64(abs(th) / abs(sin(abs(kx)))));
	else
		tmp = Float64(Float64(ky * sin(abs(th))) / abs(kx));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 1.35e-24)
		tmp = ky * (abs(th) / abs(sin(abs(kx))));
	else
		tmp = (ky * sin(abs(th))) / abs(kx);
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 1.35e-24], N[(ky * N[(N[Abs[th], $MachinePrecision] / N[Abs[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky * N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 1.35000000000000003e-24

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.5

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{ky \cdot th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 18.8%

      \[\leadsto \frac{ky \cdot th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{ky \cdot th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{ky \cdot th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. associate-/l* N/A

         \[\leadsto ky \cdot \color{blue}{\frac{th}{\sqrt{{\sin kx}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto ky \cdot \color{blue}{\frac{th}{\sqrt{{\sin kx}^{2}}}} \]

      5. lower-/.f64 19.7

         \[\leadsto ky \cdot \frac{th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      7. lift-pow.f64 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{{\sin kx}^{2}}} \]

      8. pow2 N/A

         \[\leadsto ky \cdot \frac{th}{\sqrt{\sin kx \cdot \sin kx}} \]

      9. rem-sqrt-square-rev N/A

         \[\leadsto ky \cdot \frac{th}{\left|\sin kx\right|} \]

      10. lower-fabs.f64 21.1

          \[\leadsto ky \cdot \frac{th}{\left|\sin kx\right|} \]

   9. Applied rewrites 21.1%

      \[\leadsto ky \cdot \color{blue}{\frac{th}{\left|\sin kx\right|}} \]

   if 1.35000000000000003e-24 < th

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

      6. lower-sin.f64 35.5

         \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   4. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{kx} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{kx} \]

      3. lower-sin.f64 16.0

         \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   7. Applied rewrites 16.0%

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 21.6% accurate, 4.2× speedup

Math

\[ky \cdot \frac{\sin th}{\left|kx\right|} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* ky (/ (sin th) (fabs kx))))

C

double code(double kx, double ky, double th) {
	return ky * (sin(th) / fabs(kx));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = ky * (sin(th) / abs(kx))
end function

Java

public static double code(double kx, double ky, double th) {
	return ky * (Math.sin(th) / Math.abs(kx));
}

Python

def code(kx, ky, th):
	return ky * (math.sin(th) / math.fabs(kx))

Julia

function code(kx, ky, th)
	return Float64(ky * Float64(sin(th) / abs(kx)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = ky * (sin(th) / abs(kx));
end

Wolfram

code[kx_, ky_, th_] := N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in ky around 0

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   6. lower-sin.f64 35.5

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

4. Applied rewrites 35.5%

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   3. lower-sin.f64 16.0

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

7. Applied rewrites 16.0%

   \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   3. associate-/l* N/A

      \[\leadsto ky \cdot \frac{\sin th}{\color{blue}{kx}} \]

   4. lower-*.f64 N/A

      \[\leadsto ky \cdot \frac{\sin th}{\color{blue}{kx}} \]

   5. lower-/.f64 16.8

      \[\leadsto ky \cdot \frac{\sin th}{kx} \]

9. Applied rewrites 16.8%

   \[\leadsto ky \cdot \frac{\sin th}{\color{blue}{kx}} \]
10. Add Preprocessing

Alternative 22: 15.5% accurate, 20.0× speedup

Math

\[th \cdot \frac{ky}{\left|kx\right|} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* th (/ ky (fabs kx))))

C

double code(double kx, double ky, double th) {
	return th * (ky / fabs(kx));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th * (ky / abs(kx))
end function

Java

public static double code(double kx, double ky, double th) {
	return th * (ky / Math.abs(kx));
}

Python

def code(kx, ky, th):
	return th * (ky / math.fabs(kx))

Julia

function code(kx, ky, th)
	return Float64(th * Float64(ky / abs(kx)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th * (ky / abs(kx));
end

Wolfram

code[kx_, ky_, th_] := N[(th * N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in ky around 0

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

   3. lower-sin.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{\color{blue}{2}}}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

   6. lower-sin.f64 35.5

      \[\leadsto \frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}} \]

4. Applied rewrites 35.5%

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]

5. Taylor expanded in kx around 0

   \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

   3. lower-sin.f64 16.0

      \[\leadsto \frac{ky \cdot \sin th}{kx} \]

7. Applied rewrites 16.0%

   \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]

8. Taylor expanded in th around 0

   \[\leadsto \frac{ky \cdot th}{kx} \]
9. Step-by-step derivation

10. Applied rewrites 12.8%

    \[\leadsto \frac{ky \cdot th}{kx} \]
11. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{ky \cdot th}{kx} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{ky \cdot th}{kx} \]

    3. *-commutative N/A

       \[\leadsto \frac{th \cdot ky}{kx} \]

    4. associate-/l* N/A

       \[\leadsto th \cdot \frac{ky}{\color{blue}{kx}} \]

    5. lower-*.f64 N/A

       \[\leadsto th \cdot \frac{ky}{\color{blue}{kx}} \]

    6. lower-/.f64 13.6

       \[\leadsto th \cdot \frac{ky}{kx} \]

12. Applied rewrites 13.6%

    \[\leadsto th \cdot \frac{ky}{\color{blue}{kx}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025171 
(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)))