Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 12.0s

Alternatives: 26

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   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

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

4. Applied rewrites 99.7%

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

Alternative 2: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_1 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 0.2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1 + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.999:\\ \;\;\;\;\frac{\sin ky}{t\_2} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{t\_2} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (hypot (sin ky) (sin kx)))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_1 t_3)))))
   (if (<= t_4 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.1)
       (/ (* (sin ky) th) t_2)
       (if (<= t_4 0.2)
         (* (/ (sin ky) (sqrt (+ t_1 (* ky ky)))) (sin th))
         (if (<= t_4 0.999)
           (* (/ (sin ky) t_2) (* (fma (* th th) -0.16666666666666666 1.0) th))
           (if (<= t_4 2.0)
             (*
              (/
               (sin ky)
               (sqrt
                (+ (* (* (fma -0.3333333333333333 (* kx kx) 1.0) kx) kx) t_3)))
              (sin th))
             (*
              (/ (sin th) t_2)
              (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = hypot(sin(ky), sin(kx));
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_1 + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = (sin(ky) * th) / t_2;
	} else if (t_4 <= 0.2) {
		tmp = (sin(ky) / sqrt((t_1 + (ky * ky)))) * sin(th);
	} else if (t_4 <= 0.999) {
		tmp = (sin(ky) / t_2) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_4 <= 2.0) {
		tmp = (sin(ky) / sqrt((((fma(-0.3333333333333333, (kx * kx), 1.0) * kx) * kx) + t_3))) * sin(th);
	} else {
		tmp = (sin(th) / t_2) * (fma((ky * ky), -0.16666666666666666, 1.0) * ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = hypot(sin(ky), sin(kx))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_1 + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = Float64(Float64(sin(ky) * th) / t_2);
	elseif (t_4 <= 0.2)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(t_1 + Float64(ky * ky)))) * sin(th));
	elseif (t_4 <= 0.999)
		tmp = Float64(Float64(sin(ky) / t_2) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(Float64(fma(-0.3333333333333333, Float64(kx * kx), 1.0) * kx) * kx) + t_3))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) / t_2) * Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.999], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(-0.3333333333333333 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 83.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

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

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 46.0

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

   7. Applied rewrites 46.0%

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

   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))))) < 0.20000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

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

   1. Initial program 99.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 35.3

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

   7. Applied rewrites 35.3%

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

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

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 3: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ t_5 := \frac{\sin ky}{\sqrt{t\_1 + t\_3}}\\ \mathbf{if}\;t\_5 \leq -1:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq -0.1:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_2}\\ \mathbf{elif}\;t\_5 \leq 0.2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1 + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;t\_5 \leq 0.998:\\ \;\;\;\;\frac{\sin ky}{t\_2} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{t\_2} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (hypot (sin ky) (sin kx)))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th)))
        (t_5 (/ (sin ky) (sqrt (+ t_1 t_3)))))
   (if (<= t_5 -1.0)
     t_4
     (if (<= t_5 -0.1)
       (/ (* (sin ky) th) t_2)
       (if (<= t_5 0.2)
         (* (/ (sin ky) (sqrt (+ t_1 (* ky ky)))) (sin th))
         (if (<= t_5 0.998)
           (* (/ (sin ky) t_2) (* (fma (* th th) -0.16666666666666666 1.0) th))
           (if (<= t_5 2.0)
             t_4
             (*
              (/ (sin th) t_2)
              (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = hypot(sin(ky), sin(kx));
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	double t_5 = sin(ky) / sqrt((t_1 + t_3));
	double tmp;
	if (t_5 <= -1.0) {
		tmp = t_4;
	} else if (t_5 <= -0.1) {
		tmp = (sin(ky) * th) / t_2;
	} else if (t_5 <= 0.2) {
		tmp = (sin(ky) / sqrt((t_1 + (ky * ky)))) * sin(th);
	} else if (t_5 <= 0.998) {
		tmp = (sin(ky) / t_2) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_5 <= 2.0) {
		tmp = t_4;
	} else {
		tmp = (sin(th) / t_2) * (fma((ky * ky), -0.16666666666666666, 1.0) * ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = hypot(sin(ky), sin(kx))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th))
	t_5 = Float64(sin(ky) / sqrt(Float64(t_1 + t_3)))
	tmp = 0.0
	if (t_5 <= -1.0)
		tmp = t_4;
	elseif (t_5 <= -0.1)
		tmp = Float64(Float64(sin(ky) * th) / t_2);
	elseif (t_5 <= 0.2)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(t_1 + Float64(ky * ky)))) * sin(th));
	elseif (t_5 <= 0.998)
		tmp = Float64(Float64(sin(ky) / t_2) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_5 <= 2.0)
		tmp = t_4;
	else
		tmp = Float64(Float64(sin(th) / t_2) * Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -1.0], t$95$4, If[LessEqual[t$95$5, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.998], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.0], t$95$4, N[(N[(N[Sin[th], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1 or 0.998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 92.1%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

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

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 46.0

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

   7. Applied rewrites 46.0%

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

   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))))) < 0.20000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 99.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.8

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

   7. Applied rewrites 36.8%

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

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

   1. Initial program 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 4: 83.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{kx \cdot kx + t\_2}} \cdot \sin th\\ t_4 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.998:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{t\_1} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (* (/ (sin ky) (sqrt (+ (* kx kx) t_2))) (sin th)))
        (t_4 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_4 -1.0)
     t_3
     (if (<= t_4 -0.1)
       (/ (* (sin ky) th) t_1)
       (if (<= t_4 0.2)
         (*
          (/
           (sin ky)
           (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
          (sin th))
         (if (<= t_4 0.998)
           (* (/ (sin ky) t_1) (* (fma (* th th) -0.16666666666666666 1.0) th))
           (if (<= t_4 2.0)
             t_3
             (*
              (/ (sin th) t_1)
              (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	double t_4 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = t_3;
	} else if (t_4 <= -0.1) {
		tmp = (sin(ky) * th) / t_1;
	} else if (t_4 <= 0.2) {
		tmp = (sin(ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else if (t_4 <= 0.998) {
		tmp = (sin(ky) / t_1) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_4 <= 2.0) {
		tmp = t_3;
	} else {
		tmp = (sin(th) / t_1) * (fma((ky * ky), -0.16666666666666666, 1.0) * ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_2))) * sin(th))
	t_4 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = t_3;
	elseif (t_4 <= -0.1)
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	elseif (t_4 <= 0.2)
		tmp = Float64(Float64(sin(ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	elseif (t_4 <= 0.998)
		tmp = Float64(Float64(sin(ky) / t_1) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_4 <= 2.0)
		tmp = t_3;
	else
		tmp = Float64(Float64(sin(th) / t_1) * Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], t$95$3, If[LessEqual[t$95$4, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.998], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], t$95$3, N[(N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1 or 0.998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 92.1%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

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

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 46.0

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

   7. Applied rewrites 46.0%

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

   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))))) < 0.20000000000000001

   1. Initial program 99.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 96.8

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

   7. Applied rewrites 96.8%

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

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 99.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.8

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

   7. Applied rewrites 36.8%

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

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

   1. Initial program 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 5: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.998:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{kx \cdot kx}{t\_2}, 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{t\_1} \cdot \left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -1.0)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (sin th))
     (if (<= t_3 -0.1)
       (/ (* (sin ky) th) t_1)
       (if (<= t_3 0.2)
         (*
          (/
           (sin ky)
           (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
          (sin th))
         (if (<= t_3 0.998)
           (* (/ (sin ky) t_1) (* (fma (* th th) -0.16666666666666666 1.0) th))
           (if (<= t_3 2.0)
             (/ (sin th) (fma (/ (* kx kx) t_2) 0.5 1.0))
             (*
              (/ (sin th) t_1)
              (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (sin(ky) * th) / t_1;
	} else if (t_3 <= 0.2) {
		tmp = (sin(ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else if (t_3 <= 0.998) {
		tmp = (sin(ky) / t_1) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_3 <= 2.0) {
		tmp = sin(th) / fma(((kx * kx) / t_2), 0.5, 1.0);
	} else {
		tmp = (sin(th) / t_1) * (fma((ky * ky), -0.16666666666666666, 1.0) * ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	elseif (t_3 <= 0.2)
		tmp = Float64(Float64(sin(ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	elseif (t_3 <= 0.998)
		tmp = Float64(Float64(sin(ky) / t_1) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_3 <= 2.0)
		tmp = Float64(sin(th) / fma(Float64(Float64(kx * kx) / t_2), 0.5, 1.0));
	else
		tmp = Float64(Float64(sin(th) / t_1) * Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.998], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(kx * kx), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

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

   1. Initial program 83.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

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

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 46.0

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

   7. Applied rewrites 46.0%

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

   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))))) < 0.20000000000000001

   1. Initial program 99.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 96.8

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

   7. Applied rewrites 96.8%

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

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 99.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.8

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

   7. Applied rewrites 36.8%

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

   if 0.998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\color{blue}{1 + \frac{1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 97.8

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

   7. Applied rewrites 97.8%

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

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

   1. Initial program 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 6: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ t_4 := \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;\frac{\sin ky \cdot th}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.998:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{kx \cdot kx}{t\_2}, 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2))))
        (t_4
         (*
          (/
           (sin ky)
           (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
          (sin th))))
   (if (<= t_3 -1.0)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (sin th))
     (if (<= t_3 -0.1)
       (/ (* (sin ky) th) t_1)
       (if (<= t_3 0.2)
         t_4
         (if (<= t_3 0.998)
           (* (/ (sin ky) t_1) (* (fma (* th th) -0.16666666666666666 1.0) th))
           (if (<= t_3 2.0)
             (/ (sin th) (fma (/ (* kx kx) t_2) 0.5 1.0))
             t_4)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double t_4 = (sin(ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (sin(ky) * th) / t_1;
	} else if (t_3 <= 0.2) {
		tmp = t_4;
	} else if (t_3 <= 0.998) {
		tmp = (sin(ky) / t_1) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_3 <= 2.0) {
		tmp = sin(th) / fma(((kx * kx) / t_2), 0.5, 1.0);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	t_4 = Float64(Float64(sin(ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(sin(ky) * th) / t_1);
	elseif (t_3 <= 0.2)
		tmp = t_4;
	elseif (t_3 <= 0.998)
		tmp = Float64(Float64(sin(ky) / t_1) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_3 <= 2.0)
		tmp = Float64(sin(th) / fma(Float64(Float64(kx * kx) / t_2), 0.5, 1.0));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.2], t$95$4, If[LessEqual[t$95$3, 0.998], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(kx * kx), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 83.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

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

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 46.0

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

   7. Applied rewrites 46.0%

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

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

   1. Initial program 93.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 96.9

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

   7. Applied rewrites 96.9%

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

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998

   1. Initial program 99.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.8

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

   7. Applied rewrites 36.8%

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

   if 0.998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\color{blue}{1 + \frac{1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 97.8

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

   7. Applied rewrites 97.8%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{fma}\left(\frac{kx \cdot kx}{{\sin ky}^{2}}, 0.5, 1\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ t_4 := \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.999:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{kx \cdot kx}{t\_1}, 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (* (sin ky) th) (hypot (sin ky) (sin kx))))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1))))
        (t_4
         (*
          (/
           (sin ky)
           (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
          (sin th))))
   (if (<= t_3 -1.0)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (sin th))
     (if (<= t_3 -0.1)
       t_2
       (if (<= t_3 0.2)
         t_4
         (if (<= t_3 0.999)
           t_2
           (if (<= t_3 2.0)
             (/ (sin th) (fma (/ (* kx kx) t_1) 0.5 1.0))
             t_4)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double t_4 = (sin(ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = t_2;
	} else if (t_3 <= 0.2) {
		tmp = t_4;
	} else if (t_3 <= 0.999) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = sin(th) / fma(((kx * kx) / t_1), 0.5, 1.0);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	t_4 = Float64(Float64(sin(ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = t_2;
	elseif (t_3 <= 0.2)
		tmp = t_4;
	elseif (t_3 <= 0.999)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = Float64(sin(th) / fma(Float64(Float64(kx * kx) / t_1), 0.5, 1.0));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], t$95$2, If[LessEqual[t$95$3, 0.2], t$95$4, If[LessEqual[t$95$3, 0.999], t$95$2, If[LessEqual[t$95$3, 2.0], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(kx * kx), $MachinePrecision] / t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 83.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 41.8

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

   7. Applied rewrites 41.8%

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

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

   1. Initial program 93.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 96.9

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

   7. Applied rewrites 96.9%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\color{blue}{1 + \frac{1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{fma}\left(\frac{kx \cdot kx}{{\sin ky}^{2}}, 0.5, 1\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ t_4 := \frac{\sin ky \cdot th}{t\_1}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;\frac{\sin th \cdot ky}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 0.999:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{kx \cdot kx}{t\_2}, 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin ky) (sin kx)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2))))
        (t_4 (/ (* (sin ky) th) t_1)))
   (if (<= t_3 -1.0)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (sin th))
     (if (<= t_3 -0.1)
       t_4
       (if (<= t_3 0.2)
         (/ (* (sin th) ky) t_1)
         (if (<= t_3 0.999)
           t_4
           (if (<= t_3 2.0)
             (/ (sin th) (fma (/ (* kx kx) t_2) 0.5 1.0))
             (* (/ ky kx) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(ky), sin(kx));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double t_4 = (sin(ky) * th) / t_1;
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = t_4;
	} else if (t_3 <= 0.2) {
		tmp = (sin(th) * ky) / t_1;
	} else if (t_3 <= 0.999) {
		tmp = t_4;
	} else if (t_3 <= 2.0) {
		tmp = sin(th) / fma(((kx * kx) / t_2), 0.5, 1.0);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	t_4 = Float64(Float64(sin(ky) * th) / t_1)
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = t_4;
	elseif (t_3 <= 0.2)
		tmp = Float64(Float64(sin(th) * ky) / t_1);
	elseif (t_3 <= 0.999)
		tmp = t_4;
	elseif (t_3 <= 2.0)
		tmp = Float64(sin(th) / fma(Float64(Float64(kx * kx) / t_2), 0.5, 1.0));
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], t$95$4, If[LessEqual[t$95$3, 0.2], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.999], t$95$4, If[LessEqual[t$95$3, 2.0], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(kx * kx), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 83.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 41.8

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

   7. Applied rewrites 41.8%

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

   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))))) < 0.20000000000000001

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 94.0

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

   7. Applied rewrites 94.0%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\color{blue}{1 + \frac{1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 2.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 2.4

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 5 regimes

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

   1. Initial program 83.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 87.4

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 65.2

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

   7. Applied rewrites 65.2%

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

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

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 35.5

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

   7. Applied rewrites 35.5%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 100.0

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\color{blue}{1 + \frac{1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 100.0

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 2.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 2.4

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(2 \cdot ky\right)\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - t\_1 \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.999:\\ \;\;\;\;\frac{\sin ky \cdot th}{\frac{\sqrt{\mathsf{fma}\left(1 - t\_1, 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{\sin th}{\mathsf{fma}\left(\frac{kx \cdot kx}{t\_2}, 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* 2.0 ky)))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* t_1 0.5))))) (sin th))
     (if (<= t_3 0.2)
       (* (/ (sin ky) (sqrt (- 0.5 (* (cos (* -2.0 kx)) 0.5)))) (sin th))
       (if (<= t_3 0.999)
         (/
          (* (sin ky) th)
          (/
           (sqrt (fma (- 1.0 t_1) 2.0 (* 2.0 (- 1.0 (cos (* 2.0 kx))))))
           2.0))
         (if (<= t_3 2.0)
           (/ (sin th) (fma (/ (* kx kx) t_2) 0.5 1.0))
           (* (/ ky kx) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((2.0 * ky));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (t_1 * 0.5))))) * sin(th);
	} else if (t_3 <= 0.2) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((-2.0 * kx)) * 0.5)))) * sin(th);
	} else if (t_3 <= 0.999) {
		tmp = (sin(ky) * th) / (sqrt(fma((1.0 - t_1), 2.0, (2.0 * (1.0 - cos((2.0 * kx)))))) / 2.0);
	} else if (t_3 <= 2.0) {
		tmp = sin(th) / fma(((kx * kx) / t_2), 0.5, 1.0);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(2.0 * ky))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(t_1 * 0.5))))) * sin(th));
	elseif (t_3 <= 0.2)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(-2.0 * kx)) * 0.5)))) * sin(th));
	elseif (t_3 <= 0.999)
		tmp = Float64(Float64(sin(ky) * th) / Float64(sqrt(fma(Float64(1.0 - t_1), 2.0, Float64(2.0 * Float64(1.0 - cos(Float64(2.0 * kx)))))) / 2.0));
	elseif (t_3 <= 2.0)
		tmp = Float64(sin(th) / fma(Float64(Float64(kx * kx) / t_2), 0.5, 1.0));
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.2], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.999], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] * 2.0 + N[(2.0 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(kx * kx), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 83.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 83.7

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 83.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky + ky\right)}\right)}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 66.8

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   7. Applied rewrites 66.8%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 87.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 87.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      5. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      9. metadata-eval 65.2

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot \sin th \]

   7. Applied rewrites 65.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}}} \cdot \sin th \]

   if 0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lower-*.f64 99.4

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}} \]

      14. lower-hypot.f64 99.5

          \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \sin ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

      3. lower-sin.f64 35.5

         \[\leadsto \frac{\color{blue}{\sin ky} \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

   7. Applied rewrites 35.5%

      \[\leadsto \frac{\color{blue}{\sin ky \cdot th}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
   8. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky \cdot th}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}} \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \]

      8. frac-add N/A

         \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \]

      10. sqrt-div N/A

          \[\leadsto \frac{\sin ky \cdot th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{4}}}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\sin ky \cdot th}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\color{blue}{2}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{2}}} \]

   9. Applied rewrites 35.5%

      \[\leadsto \frac{\sin ky \cdot th}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \]

   if 0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 100.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. clear-num N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      7. lower-/.f64 100.0

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}}{\sin ky}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{{\sin ky}^{2} + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{{\sin ky}^{2}} + \sin kx \cdot \sin kx}}{\sin ky}} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + \sin kx \cdot \sin kx}}{\sin ky}} \]

      15. lower-hypot.f64 100.0

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\color{blue}{1 + \frac{1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{1}{2} \cdot \frac{{kx}^{2}}{{\sin ky}^{2}} + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{{kx}^{2}}{{\sin ky}^{2}} \cdot \frac{1}{2}} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{fma}\left(\frac{{kx}^{2}}{{\sin ky}^{2}}, \frac{1}{2}, 1\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{fma}\left(\color{blue}{\frac{{kx}^{2}}{{\sin ky}^{2}}}, \frac{1}{2}, 1\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th}{\mathsf{fma}\left(\frac{\color{blue}{kx \cdot kx}}{{\sin ky}^{2}}, \frac{1}{2}, 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{fma}\left(\frac{\color{blue}{kx \cdot kx}}{{\sin ky}^{2}}, \frac{1}{2}, 1\right)} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{fma}\left(\frac{kx \cdot kx}{\color{blue}{{\sin ky}^{2}}}, \frac{1}{2}, 1\right)} \]

      8. lower-sin.f64 100.0

         \[\leadsto \frac{\sin th}{\mathsf{fma}\left(\frac{kx \cdot kx}{{\color{blue}{\sin ky}}^{2}}, 0.5, 1\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{fma}\left(\frac{kx \cdot kx}{{\sin ky}^{2}}, 0.5, 1\right)}} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 34.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(ky \cdot th\right) \cdot \sqrt{{\left(0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5\right)}^{-1}}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\left({kx}^{-1} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (* ky th) (sqrt (pow (- 0.5 (* (cos (* -2.0 kx)) 0.5)) -1.0))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 2e-155)
     t_1
     (if (<= t_2 2e-87)
       (* (* (pow kx -1.0) ky) (sin th))
       (if (<= t_2 5e-10)
         t_1
         (if (<= t_2 2.0) (sin th) (* (/ ky kx) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (ky * th) * sqrt(pow((0.5 - (cos((-2.0 * kx)) * 0.5)), -1.0));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 2e-155) {
		tmp = t_1;
	} else if (t_2 <= 2e-87) {
		tmp = (pow(kx, -1.0) * ky) * sin(th);
	} else if (t_2 <= 5e-10) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (ky * th) * sqrt(((0.5d0 - (cos(((-2.0d0) * kx)) * 0.5d0)) ** (-1.0d0)))
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_2 <= 2d-155) then
        tmp = t_1
    else if (t_2 <= 2d-87) then
        tmp = ((kx ** (-1.0d0)) * ky) * sin(th)
    else if (t_2 <= 5d-10) then
        tmp = t_1
    else if (t_2 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = (ky / kx) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (ky * th) * Math.sqrt(Math.pow((0.5 - (Math.cos((-2.0 * kx)) * 0.5)), -1.0));
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= 2e-155) {
		tmp = t_1;
	} else if (t_2 <= 2e-87) {
		tmp = (Math.pow(kx, -1.0) * ky) * Math.sin(th);
	} else if (t_2 <= 5e-10) {
		tmp = t_1;
	} else if (t_2 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / kx) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (ky * th) * math.sqrt(math.pow((0.5 - (math.cos((-2.0 * kx)) * 0.5)), -1.0))
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= 2e-155:
		tmp = t_1
	elif t_2 <= 2e-87:
		tmp = (math.pow(kx, -1.0) * ky) * math.sin(th)
	elif t_2 <= 5e-10:
		tmp = t_1
	elif t_2 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / kx) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(ky * th) * sqrt((Float64(0.5 - Float64(cos(Float64(-2.0 * kx)) * 0.5)) ^ -1.0)))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= 2e-155)
		tmp = t_1;
	elseif (t_2 <= 2e-87)
		tmp = Float64(Float64((kx ^ -1.0) * ky) * sin(th));
	elseif (t_2 <= 5e-10)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (ky * th) * sqrt(((0.5 - (cos((-2.0 * kx)) * 0.5)) ^ -1.0));
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= 2e-155)
		tmp = t_1;
	elseif (t_2 <= 2e-87)
		tmp = ((kx ^ -1.0) * ky) * sin(th);
	elseif (t_2 <= 5e-10)
		tmp = t_1;
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = (ky / kx) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(ky * th), $MachinePrecision] * N[Sqrt[N[Power[N[(0.5 - N[(N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 2e-155], t$95$1, If[LessEqual[t$95$2, 2e-87], N[(N[(N[Power[kx, -1.0], $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-10], t$95$1, If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000003e-155 or 2.00000000000000004e-87 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.00000000000000031e-10

   1. Initial program 95.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 89.3

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 89.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sin th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sin th\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sin th\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sin th\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sin th\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \left(ky \cdot \sin th\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \left(ky \cdot \sin th\right) \]

      8. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \left(-2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot \color{blue}{\left(\sin th \cdot ky\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \left(-2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot \color{blue}{\left(\sin th \cdot ky\right)} \]

      15. lower-sin.f64 47.6

          \[\leadsto \sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \left(\color{blue}{\sin th} \cdot ky\right) \]

   7. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \left(\sin th \cdot ky\right)} \]

   8. Taylor expanded in th around 0

      \[\leadsto \left(ky \cdot th\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-2 \cdot kx\right)}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 25.2%

       \[\leadsto \left(ky \cdot th\right) \cdot \color{blue}{\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}}} \]

   if 2.00000000000000003e-155 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000004e-87

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 52.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 52.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 50.8

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\frac{1}{kx} \cdot ky\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 52.0%

       \[\leadsto \left(\frac{1}{kx} \cdot ky\right) \cdot \sin th \]

   if 5.00000000000000031e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 69.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\left(ky \cdot th\right) \cdot \sqrt{{\left(0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\left({kx}^{-1} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\left(ky \cdot th\right) \cdot \sqrt{{\left(0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -1.0)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (sin th))
     (if (<= t_1 0.7)
       (* (/ (sin ky) (sqrt (- 0.5 (* (cos (* -2.0 kx)) 0.5)))) (sin th))
       (if (<= t_1 2.0) (sin th) (* (/ ky kx) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	} else if (t_1 <= 0.7) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((-2.0 * kx)) * 0.5)))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-1.0d0)) then
        tmp = (sin(ky) / sqrt(((kx * kx) + (0.5d0 - (cos((2.0d0 * ky)) * 0.5d0))))) * sin(th)
    else if (t_1 <= 0.7d0) then
        tmp = (sin(ky) / sqrt((0.5d0 - (cos(((-2.0d0) * kx)) * 0.5d0)))) * sin(th)
    else if (t_1 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = (ky / kx) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + (0.5 - (Math.cos((2.0 * ky)) * 0.5))))) * Math.sin(th);
	} else if (t_1 <= 0.7) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((-2.0 * kx)) * 0.5)))) * Math.sin(th);
	} else if (t_1 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / kx) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -1.0:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + (0.5 - (math.cos((2.0 * ky)) * 0.5))))) * math.sin(th)
	elif t_1 <= 0.7:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((-2.0 * kx)) * 0.5)))) * math.sin(th)
	elif t_1 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / kx) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * sin(th));
	elseif (t_1 <= 0.7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(-2.0 * kx)) * 0.5)))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	elseif (t_1 <= 0.7)
		tmp = (sin(ky) / sqrt((0.5 - (cos((-2.0 * kx)) * 0.5)))) * sin(th);
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = (ky / kx) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 83.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 83.7

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 83.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky + ky\right)}\right)}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 66.8

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   7. Applied rewrites 66.8%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 88.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 88.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      5. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      9. metadata-eval 63.1

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot \sin th \]

   7. Applied rewrites 63.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}}} \cdot \sin th \]

   if 0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 78.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 54.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_1 \leq 0.7:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -1.0)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (* (fma (* th th) -0.16666666666666666 1.0) th))
     (if (<= t_1 0.7)
       (* (/ (sin ky) (sqrt (- 0.5 (* (cos (* -2.0 kx)) 0.5)))) (sin th))
       (if (<= t_1 2.0) (sin th) (* (/ ky kx) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_1 <= 0.7) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((-2.0 * kx)) * 0.5)))) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_1 <= 0.7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(-2.0 * kx)) * 0.5)))) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 83.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 83.7

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 83.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th\right) \]

      7. lower-*.f64 48.8

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th\right) \]

   8. Applied rewrites 48.8%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky + ky\right)}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      12. lower-*.f64 36.7

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   10. Applied rewrites 36.7%

       \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.69999999999999996

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 88.0

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 88.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \sin th \]

      3. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \sin th \]

      5. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \sin th \]

      9. metadata-eval 63.1

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot \sin th \]

   7. Applied rewrites 63.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}}} \cdot \sin th \]

   if 0.69999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 78.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 78.5%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\left(\sqrt{{\left(\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)\right)}^{-1}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (*
      (/ (sin ky) (sqrt (+ (* kx kx) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (* (fma (* th th) -0.16666666666666666 1.0) th))
     (if (<= t_1 1e-9)
       (* (* (sqrt (pow (fma (cos (* 2.0 kx)) -0.5 0.5) -1.0)) ky) (sin th))
       (if (<= t_1 2.0) (sin th) (* (/ ky kx) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_1 <= 1e-9) {
		tmp = (sqrt(pow(fma(cos((2.0 * kx)), -0.5, 0.5), -1.0)) * ky) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_1 <= 1e-9)
		tmp = Float64(Float64(sqrt((fma(cos(Float64(2.0 * kx)), -0.5, 0.5) ^ -1.0)) * ky) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-9], N[(N[(N[Sqrt[N[Power[N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 49.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 49.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th\right) \]

      7. lower-*.f64 29.2

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th\right) \]

   8. Applied rewrites 29.2%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky + ky\right)}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      12. lower-*.f64 21.5

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   10. Applied rewrites 21.5%

       \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000006e-9

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 83.8

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 83.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 83.6

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 83.6%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot ky\right) \cdot \sin th \]

   if 1.00000000000000006e-9 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 69.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-9}:\\ \;\;\;\;\left(\sqrt{{\left(\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)\right)}^{-1}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 0.0005 \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (or (<= t_1 0.0005) (not (<= t_1 2.0)))
     (* (/ ky kx) (sin th))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if ((t_1 <= 0.0005) || !(t_1 <= 2.0)) {
		tmp = (ky / kx) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if ((t_1 <= 0.0005d0) .or. (.not. (t_1 <= 2.0d0))) then
        tmp = (ky / kx) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if ((t_1 <= 0.0005) || !(t_1 <= 2.0)) {
		tmp = (ky / kx) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if (t_1 <= 0.0005) or not (t_1 <= 2.0):
		tmp = (ky / kx) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if ((t_1 <= 0.0005) || !(t_1 <= 2.0))
		tmp = Float64(Float64(ky / kx) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if ((t_1 <= 0.0005) || ~((t_1 <= 2.0)))
		tmp = (ky / kx) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.0005], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-4 or 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 83.2

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 83.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 45.9

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 21.6%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]

   if 5.0000000000000001e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 70.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0005 \lor \neg \left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2\right):\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\left(\sqrt{{\left(\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)\right)}^{-1}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 1e-9)
     (* (* (sqrt (pow (fma (cos (* 2.0 kx)) -0.5 0.5) -1.0)) ky) (sin th))
     (if (<= t_1 2.0) (sin th) (* (/ ky kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 1e-9) {
		tmp = (sqrt(pow(fma(cos((2.0 * kx)), -0.5, 0.5), -1.0)) * ky) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 1e-9)
		tmp = Float64(Float64(sqrt((fma(cos(Float64(2.0 * kx)), -0.5, 0.5) ^ -1.0)) * ky) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-9], N[(N[(N[Sqrt[N[Power[N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000006e-9

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 86.8

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 86.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 47.9

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]
   8. Step-by-step derivation

   9. Applied rewrites 47.9%

      \[\leadsto \left(\sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot ky\right) \cdot \sin th \]

   if 1.00000000000000006e-9 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 69.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-9}:\\ \;\;\;\;\left(\sqrt{{\left(\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)\right)}^{-1}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 10^{-9}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)\right)}^{-1}} \cdot \left(\sin th \cdot ky\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 1e-9)
     (* (sqrt (pow (fma (cos (* 2.0 kx)) -0.5 0.5) -1.0)) (* (sin th) ky))
     (if (<= t_1 2.0) (sin th) (* (/ ky kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 1e-9) {
		tmp = sqrt(pow(fma(cos((2.0 * kx)), -0.5, 0.5), -1.0)) * (sin(th) * ky);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 1e-9)
		tmp = Float64(sqrt((fma(cos(Float64(2.0 * kx)), -0.5, 0.5) ^ -1.0)) * Float64(sin(th) * ky));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-9], N[(N[Sqrt[N[Power[N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000006e-9

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 86.8

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 86.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sin th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sin th\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sin th\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sin th\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sin th\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \left(ky \cdot \sin th\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \left(ky \cdot \sin th\right) \]

      8. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \left(-2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot \color{blue}{\left(\sin th \cdot ky\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \left(-2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot \color{blue}{\left(\sin th \cdot ky\right)} \]

      15. lower-sin.f64 47.8

          \[\leadsto \sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \left(\color{blue}{\sin th} \cdot ky\right) \]

   7. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \left(\sin th \cdot ky\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.8%

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)}} \cdot \left(\sin th \cdot ky\right) \]

   if 1.00000000000000006e-9 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 69.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-9}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(\cos \left(2 \cdot kx\right), -0.5, 0.5\right)\right)}^{-1}} \cdot \left(\sin th \cdot ky\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 0.0005:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 0.0005)
     (* (/ ky (sin kx)) (sin th))
     (if (<= t_1 2.0) (sin th) (* (/ ky kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 0.0005) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= 0.0005d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else if (t_1 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = (ky / kx) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 0.0005) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else if (t_1 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / kx) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 0.0005:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	elif t_1 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / kx) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 0.0005)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 0.0005)
		tmp = (ky / sin(kx)) * sin(th);
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = (ky / kx) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0005], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-4

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 37.0

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 37.0%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 5.0000000000000001e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 70.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 42.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 6 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 6e-13)
     (/ (* (sin th) ky) (sin kx))
     (if (<= t_1 2.0) (sin th) (* (/ ky kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 6e-13) {
		tmp = (sin(th) * ky) / sin(kx);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= 6d-13) then
        tmp = (sin(th) * ky) / sin(kx)
    else if (t_1 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = (ky / kx) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 6e-13) {
		tmp = (Math.sin(th) * ky) / Math.sin(kx);
	} else if (t_1 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / kx) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 6e-13:
		tmp = (math.sin(th) * ky) / math.sin(kx)
	elif t_1 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / kx) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 6e-13)
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 6e-13)
		tmp = (sin(th) * ky) / sin(kx);
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = (ky / kx) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 6e-13], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.99999999999999968e-13

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      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

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sin kx} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sin kx} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th} \cdot ky}{\sin kx} \]

      5. lower-sin.f64 35.1

         \[\leadsto \frac{\sin th \cdot ky}{\color{blue}{\sin kx}} \]

   7. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   if 5.99999999999999968e-13 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 68.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 68.4%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 6 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 0.0005:\\ \;\;\;\;\left(\sqrt{{\left(kx \cdot kx\right)}^{-1}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 0.0005)
     (* (* (sqrt (pow (* kx kx) -1.0)) ky) (sin th))
     (if (<= t_1 2.0) (sin th) (* (/ ky kx) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 0.0005) {
		tmp = (sqrt(pow((kx * kx), -1.0)) * ky) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (ky / kx) * sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= 0.0005d0) then
        tmp = (sqrt(((kx * kx) ** (-1.0d0))) * ky) * sin(th)
    else if (t_1 <= 2.0d0) then
        tmp = sin(th)
    else
        tmp = (ky / kx) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 0.0005) {
		tmp = (Math.sqrt(Math.pow((kx * kx), -1.0)) * ky) * Math.sin(th);
	} else if (t_1 <= 2.0) {
		tmp = Math.sin(th);
	} else {
		tmp = (ky / kx) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= 0.0005:
		tmp = (math.sqrt(math.pow((kx * kx), -1.0)) * ky) * math.sin(th)
	elif t_1 <= 2.0:
		tmp = math.sin(th)
	else:
		tmp = (ky / kx) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 0.0005)
		tmp = Float64(Float64(sqrt((Float64(kx * kx) ^ -1.0)) * ky) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(ky / kx) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= 0.0005)
		tmp = (sqrt(((kx * kx) ^ -1.0)) * ky) * sin(th);
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = (ky / kx) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0005], N[(N[(N[Sqrt[N[Power[N[(kx * kx), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * ky), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-4

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 86.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 86.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 47.6

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\sqrt{\frac{1}{{kx}^{2}}} \cdot ky\right) \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 22.7%

       \[\leadsto \left(\sqrt{\frac{1}{kx \cdot kx}} \cdot ky\right) \cdot \sin th \]

   if 5.0000000000000001e-4 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 70.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 2.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot ky\right)} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot ky\right) \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot ky\right) \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot ky\right) \cdot \sin th \]

      12. metadata-eval 2.4

          \[\leadsto \left(\sqrt{\frac{1}{0.5 - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot 0.5}} \cdot ky\right) \cdot \sin th \]

   7. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot ky\right)} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 44.0%

       \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.0005:\\ \;\;\;\;\left(\sqrt{{\left(kx \cdot kx\right)}^{-1}} \cdot ky\right) \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 15.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 2 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{-th}{\mathsf{fma}\left(-0.16666666666666666, th \cdot th, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      2e-308)
   (* (* (* -0.16666666666666666 th) th) th)
   (/ (- th) (fma -0.16666666666666666 (* th th) -1.0))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 2e-308) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = -th / fma(-0.16666666666666666, (th * th), -1.0);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-308)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = Float64(Float64(-th) / fma(-0.16666666666666666, Float64(th * th), -1.0));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 2e-308], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[((-th) / N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 1.9999999999999998e-308

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 24.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 24.7%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.4%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 15.3%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 15.3%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 1.9999999999999998e-308 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 94.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 20.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.8%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]
   9. Step-by-step derivation

   10. Applied rewrites 11.2%

       \[\leadsto \frac{\mathsf{fma}\left(0.027777777777777776, {th}^{4}, -1\right) \cdot th}{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{th \cdot th}, -1\right)} \]

   11. Taylor expanded in th around 0

       \[\leadsto \frac{-1 \cdot th}{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th} \cdot th, -1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 13.2%

       \[\leadsto \frac{-th}{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{th} \cdot th, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 15.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 2 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;1 \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      2e-308)
   (* (* (* -0.16666666666666666 th) th) th)
   (* 1.0 th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 2e-308) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = 1.0 * th;
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)) <= 2d-308) then
        tmp = (((-0.16666666666666666d0) * th) * th) * th
    else
        tmp = 1.0d0 * th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th)) <= 2e-308) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = 1.0 * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)) <= 2e-308:
		tmp = ((-0.16666666666666666 * th) * th) * th
	else:
		tmp = 1.0 * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-308)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = Float64(1.0 * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-308)
		tmp = ((-0.16666666666666666 * th) * th) * th;
	else
		tmp = 1.0 * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 2e-308], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[(1.0 * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 1.9999999999999998e-308

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 24.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 24.7%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.4%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 15.3%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 15.3%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 1.9999999999999998e-308 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 94.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 20.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.8%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 4.7%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   12. Taylor expanded in th around 0

       \[\leadsto 1 \cdot th \]
   13. Step-by-step derivation

   14. Applied rewrites 12.2%

       \[\leadsto 1 \cdot th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 35.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sin th \cdot ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-40)
   (/ (* (sin th) ky) kx)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-40) {
		tmp = (sin(th) * ky) / kx;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-40) then
        tmp = (sin(th) * ky) / kx
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-40) {
		tmp = (Math.sin(th) * ky) / kx;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5e-40:
		tmp = (math.sin(th) * ky) / kx
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-40)
		tmp = Float64(Float64(sin(th) * ky) / kx);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-40)
		tmp = (sin(th) * ky) / kx;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-40], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / kx), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.99999999999999965e-40

   1. Initial program 95.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 86.3

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 86.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\left(ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sin th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}} \cdot \left(ky \cdot \sin th\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sin th\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sin th\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}}} \cdot \left(ky \cdot \sin th\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \left(ky \cdot \sin th\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}}} \cdot \left(ky \cdot \sin th\right) \]

      8. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)} \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \left(\color{blue}{-2} \cdot kx\right) \cdot \frac{1}{2}}} \cdot \left(ky \cdot \sin th\right) \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \left(-2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot \color{blue}{\left(\sin th \cdot ky\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{2} - \cos \left(-2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot \color{blue}{\left(\sin th \cdot ky\right)} \]

      15. lower-sin.f64 46.0

          \[\leadsto \sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \left(\color{blue}{\sin th} \cdot ky\right) \]

   7. Applied rewrites 46.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{0.5 - \cos \left(-2 \cdot kx\right) \cdot 0.5}} \cdot \left(\sin th \cdot ky\right)} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{kx}} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.4%

       \[\leadsto \frac{\sin th \cdot ky}{\color{blue}{kx}} \]

   if 4.99999999999999965e-40 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.0

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 61.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 30.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 3.75 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
      3.75e-53)
   (* (* (* -0.16666666666666666 th) th) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 3.75e-53) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 3.75d-53) then
        tmp = (((-0.16666666666666666d0) * th) * th) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 3.75e-53) {
		tmp = ((-0.16666666666666666 * th) * th) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 3.75e-53:
		tmp = ((-0.16666666666666666 * th) * th) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 3.75e-53)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * th) * th) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 3.75e-53)
		tmp = ((-0.16666666666666666 * th) * th) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.75e-53], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 3.75e-53

   1. Initial program 95.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.4%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

   11. Applied rewrites 13.8%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 13.8%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 3.75e-53 < (/.f64 (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 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 59.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 75.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0027:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0027)
   (*
    (/
     (sin ky)
     (hypot (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (sin kx)))
    (sin th))
   (*
    (/
     (sin ky)
     (sqrt
      (+ (- 0.5 (* (cos (* 2.0 kx)) 0.5)) (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0027) {
		tmp = (sin(ky) / hypot((fma(-0.16666666666666666, (ky * ky), 1.0) * ky), sin(kx))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0027)
		tmp = Float64(Float64(sin(ky) / hypot(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky), sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0027], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0027000000000000001

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      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

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky, \sin kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky, \sin kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

      6. lower-*.f64 71.9

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky, \sin kx\right)} \cdot \sin th \]

   7. Applied rewrites 71.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}, \sin kx\right)} \cdot \sin th \]

   if 0.0027000000000000001 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + {\sin ky}^{2}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + {\sin ky}^{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(kx + kx\right)}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right) \cdot \frac{1}{2}}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + {\sin ky}^{2}}} \cdot \sin th \]

      12. lower-*.f64 99.6

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + {\sin ky}^{2}}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + {\sin ky}^{2}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky + ky\right)}\right)}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right) + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 99.4

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 13.7% accurate, 105.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* 1.0 th))

C

double code(double kx, double ky, double th) {
	return 1.0 * th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = 1.0d0 * th
end function

Java

public static double code(double kx, double ky, double th) {
	return 1.0 * th;
}

Python

def code(kx, ky, th):
	return 1.0 * th

Julia

function code(kx, ky, th)
	return Float64(1.0 * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 1.0 * th;
end

Wolfram

code[kx_, ky_, th_] := N[(1.0 * th), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 22.7

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 22.7%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 11.6%

   \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

9. Taylor expanded in th around inf

   \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
10. Step-by-step derivation

11. Applied rewrites 10.7%

    \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

12. Taylor expanded in th around 0

    \[\leadsto 1 \cdot th \]
13. Step-by-step derivation

14. Applied rewrites 11.7%

    \[\leadsto 1 \cdot th \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024305 
(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)))