Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 11.7s

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: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)}{\sin th}}\\ \mathbf{elif}\;t\_2 \leq 0.9999995735978008:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin kx) 2.0))))))
   (if (<= t_2 -0.999)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_2 -0.02)
       (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
       (if (<= t_2 1e-5)
         (/
          (sin ky)
          (/
           (hypot (sin kx) (* (fma (* ky ky) -0.16666666666666666 1.0) ky))
           (sin th)))
         (if (<= t_2 0.9999995735978008)
           (* (/ -1.0 (hypot (sin ky) (sin kx))) (* (- th) (sin ky)))
           (*
            (/
             (sin ky)
             (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) 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((t_1 + pow(sin(kx), 2.0)));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_2 <= -0.02) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (t_2 <= 1e-5) {
		tmp = sin(ky) / (hypot(sin(kx), (fma((ky * ky), -0.16666666666666666, 1.0) * ky)) / sin(th));
	} else if (t_2 <= 0.9999995735978008) {
		tmp = (-1.0 / hypot(sin(ky), sin(kx))) * (-th * sin(ky));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_2 <= -0.02)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (t_2 <= 1e-5)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)) / sin(th)));
	elseif (t_2 <= 0.9999995735978008)
		tmp = Float64(Float64(-1.0 / hypot(sin(ky), sin(kx))) * Float64(Float64(-th) * sin(ky)));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * 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[(t$95$1 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.02], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-5], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999995735978008], N[(N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 86.3

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

   5. Applied rewrites 86.3%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 59.1

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

   9. Applied rewrites 59.1%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000008e-5

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.5

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.6

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999999573597800784

   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.1

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 51.5

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

   7. Applied rewrites 51.5%

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

   if 0.999999573597800784 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right)}{\sin th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999995735978008:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;t\_3 \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9999995735978008:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_1 t_2)))))
   (if (<= t_3 -0.999)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_3 -0.02)
       (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
       (if (<= t_3 1e-5)
         (* (/ (sin ky) (sqrt (+ (* ky ky) t_2))) (sin th))
         (if (<= t_3 0.9999995735978008)
           (* (/ -1.0 (hypot (sin ky) (sin kx))) (* (- th) (sin ky)))
           (*
            (/
             (sin ky)
             (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_1 + t_2));
	double tmp;
	if (t_3 <= -0.999) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_3 <= -0.02) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (t_3 <= 1e-5) {
		tmp = (sin(ky) / sqrt(((ky * ky) + t_2))) * sin(th);
	} else if (t_3 <= 0.9999995735978008) {
		tmp = (-1.0 / hypot(sin(ky), sin(kx))) * (-th * sin(ky));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_3 <= -0.02)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (t_3 <= 1e-5)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + t_2))) * sin(th));
	elseif (t_3 <= 0.9999995735978008)
		tmp = Float64(Float64(-1.0 / hypot(sin(ky), sin(kx))) * Float64(Float64(-th) * sin(ky)));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * 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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.02], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-5], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9999995735978008], N[(N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 86.3

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

   5. Applied rewrites 86.3%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 59.1

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

   9. Applied rewrites 59.1%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000008e-5

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   if 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999999573597800784

   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.1

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 51.5

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

   7. Applied rewrites 51.5%

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

   if 0.999999573597800784 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999995735978008:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;t\_3 \leq 10^{-5}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 0.9999995735978008:\\ \;\;\;\;t\_1 \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ -1.0 (hypot (sin ky) (sin kx))))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin kx) 2.0))))))
   (if (<= t_3 -0.999)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_2))) (sin th))
     (if (<= t_3 -0.02)
       (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
       (if (<= t_3 1e-5)
         (* (* (- ky) (sin th)) t_1)
         (if (<= t_3 0.9999995735978008)
           (* t_1 (* (- th) (sin ky)))
           (*
            (/
             (sin ky)
             (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -1.0 / hypot(sin(ky), sin(kx));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(kx), 2.0)));
	double tmp;
	if (t_3 <= -0.999) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	} else if (t_3 <= -0.02) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (t_3 <= 1e-5) {
		tmp = (-ky * sin(th)) * t_1;
	} else if (t_3 <= 0.9999995735978008) {
		tmp = t_1 * (-th * sin(ky));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_2))) * sin(th));
	elseif (t_3 <= -0.02)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (t_3 <= 1e-5)
		tmp = Float64(Float64(Float64(-ky) * sin(th)) * t_1);
	elseif (t_3 <= 0.9999995735978008)
		tmp = Float64(t_1 * Float64(Float64(-th) * sin(ky)));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.02], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-5], N[(N[((-ky) * N[Sin[th], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.9999995735978008], N[(t$95$1 * N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.3%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 86.3

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

   5. Applied rewrites 86.3%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 59.1

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

   9. Applied rewrites 59.1%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000008e-5

   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 95.7

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.8

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

   7. Applied rewrites 94.8%

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

   if 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999999573597800784

   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.1

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 51.5

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

   7. Applied rewrites 51.5%

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

   if 0.999999573597800784 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-5}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999995735978008:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_3 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;t\_2 \leq -0.999:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot t\_3\\ \mathbf{elif}\;t\_2 \leq 0.9999995735978008:\\ \;\;\;\;t\_3 \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (/
           (sin ky)
           (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
          (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_3 (/ -1.0 (hypot (sin ky) (sin kx)))))
   (if (<= t_2 -0.999)
     t_1
     (if (<= t_2 -0.02)
       (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
       (if (<= t_2 1e-5)
         (* (* (- ky) (sin th)) t_3)
         (if (<= t_2 0.9999995735978008) (* t_3 (* (- th) (sin ky))) t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_3 = -1.0 / hypot(sin(ky), sin(kx));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = t_1;
	} else if (t_2 <= -0.02) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (t_2 <= 1e-5) {
		tmp = (-ky * sin(th)) * t_3;
	} else if (t_2 <= 0.9999995735978008) {
		tmp = t_3 * (-th * sin(ky));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_3 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	tmp = 0.0
	if (t_2 <= -0.999)
		tmp = t_1;
	elseif (t_2 <= -0.02)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (t_2 <= 1e-5)
		tmp = Float64(Float64(Float64(-ky) * sin(th)) * t_3);
	elseif (t_2 <= 0.9999995735978008)
		tmp = Float64(t_3 * Float64(Float64(-th) * sin(ky)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999], t$95$1, If[LessEqual[t$95$2, -0.02], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-5], N[(N[((-ky) * N[Sin[th], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 0.9999995735978008], N[(t$95$3 * N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999 or 0.999999573597800784 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 87.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 59.1

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

   9. Applied rewrites 59.1%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000008e-5

   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 95.7

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.8

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

   7. Applied rewrites 94.8%

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

   if 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999999573597800784

   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.1

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 51.5

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

   7. Applied rewrites 51.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-5}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999995735978008:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 0.9999995735978008:\\ \;\;\;\;t\_2 \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (/ -1.0 (hypot (sin ky) (sin kx)))))
   (if (<= t_1 -0.999)
     (* (/ (sin ky) (/ (sqrt (* (- 1.0 (cos (* 2.0 ky))) 2.0)) 2.0)) (sin th))
     (if (<= t_1 -0.02)
       (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
       (if (<= t_1 1e-5)
         (* (* (- ky) (sin th)) t_2)
         (if (<= t_1 0.9999995735978008)
           (* t_2 (* (- th) (sin ky)))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = -1.0 / hypot(sin(ky), sin(kx));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	} else if (t_1 <= -0.02) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (t_1 <= 1e-5) {
		tmp = (-ky * sin(th)) * t_2;
	} else if (t_1 <= 0.9999995735978008) {
		tmp = t_2 * (-th * sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_2 = -1.0 / Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (Math.sin(ky) / (Math.sqrt(((1.0 - Math.cos((2.0 * ky))) * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_1 <= -0.02) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else if (t_1 <= 1e-5) {
		tmp = (-ky * Math.sin(th)) * t_2;
	} else if (t_1 <= 0.9999995735978008) {
		tmp = t_2 * (-th * Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_2 = -1.0 / math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if t_1 <= -0.999:
		tmp = (math.sin(ky) / (math.sqrt(((1.0 - math.cos((2.0 * ky))) * 2.0)) / 2.0)) * math.sin(th)
	elif t_1 <= -0.02:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	elif t_1 <= 1e-5:
		tmp = (-ky * math.sin(th)) * t_2
	elif t_1 <= 0.9999995735978008:
		tmp = t_2 * (-th * math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 2.0)) / 2.0)) * sin(th));
	elseif (t_1 <= -0.02)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (t_1 <= 1e-5)
		tmp = Float64(Float64(Float64(-ky) * sin(th)) * t_2);
	elseif (t_1 <= 0.9999995735978008)
		tmp = Float64(t_2 * Float64(Float64(-th) * sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_2 = -1.0 / hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (t_1 <= -0.999)
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	elseif (t_1 <= -0.02)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	elseif (t_1 <= 1e-5)
		tmp = (-ky * sin(th)) * t_2;
	elseif (t_1 <= 0.9999995735978008)
		tmp = t_2 * (-th * sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-5], N[(N[((-ky) * N[Sin[th], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 0.9999995735978008], N[(t$95$2 * N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 59.1

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

   9. Applied rewrites 59.1%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000008e-5

   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 95.7

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in ky around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.8

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

   7. Applied rewrites 94.8%

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

   if 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999999573597800784

   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.1

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 51.5

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

   7. Applied rewrites 51.5%

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

   if 0.999999573597800784 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 95.1

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

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-5}:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999995735978008:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;t\_1 \leq 0.9999995735978008:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.999)
     (* (/ (sin ky) (/ (sqrt (* (- 1.0 (cos (* 2.0 ky))) 2.0)) 2.0)) (sin th))
     (if (<= t_1 -0.02)
       (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
       (if (<= t_1 1e-5)
         (/ (sin ky) (/ (sin kx) (sin th)))
         (if (<= t_1 0.9999995735978008)
           (* (/ -1.0 (hypot (sin ky) (sin kx))) (* (- th) (sin ky)))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	} else if (t_1 <= -0.02) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else if (t_1 <= 1e-5) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else if (t_1 <= 0.9999995735978008) {
		tmp = (-1.0 / hypot(sin(ky), sin(kx))) * (-th * sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (Math.sin(ky) / (Math.sqrt(((1.0 - Math.cos((2.0 * ky))) * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_1 <= -0.02) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else if (t_1 <= 1e-5) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else if (t_1 <= 0.9999995735978008) {
		tmp = (-1.0 / Math.hypot(Math.sin(ky), Math.sin(kx))) * (-th * Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.999:
		tmp = (math.sin(ky) / (math.sqrt(((1.0 - math.cos((2.0 * ky))) * 2.0)) / 2.0)) * math.sin(th)
	elif t_1 <= -0.02:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	elif t_1 <= 1e-5:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	elif t_1 <= 0.9999995735978008:
		tmp = (-1.0 / math.hypot(math.sin(ky), math.sin(kx))) * (-th * math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 2.0)) / 2.0)) * sin(th));
	elseif (t_1 <= -0.02)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	elseif (t_1 <= 1e-5)
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	elseif (t_1 <= 0.9999995735978008)
		tmp = Float64(Float64(-1.0 / hypot(sin(ky), sin(kx))) * Float64(Float64(-th) * sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.999)
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	elseif (t_1 <= -0.02)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	elseif (t_1 <= 1e-5)
		tmp = sin(ky) / (sin(kx) / sin(th));
	elseif (t_1 <= 0.9999995735978008)
		tmp = (-1.0 / hypot(sin(ky), sin(kx))) * (-th * sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-5], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999995735978008], N[(N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 59.1

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

   9. Applied rewrites 59.1%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000008e-5

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.5

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.6

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 63.8

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

   9. Applied rewrites 63.8%

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

   if 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999999573597800784

   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. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 99.1

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

      14. lift-sqrt.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 51.5

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

   7. Applied rewrites 51.5%

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

   if 0.999999573597800784 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 95.1

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

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999995735978008:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left(-th\right) \cdot \sin ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;t\_1 \leq 0.9999995735978008:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))))
   (if (<= t_1 -0.999)
     (* (/ (sin ky) (/ (sqrt (* (- 1.0 (cos (* 2.0 ky))) 2.0)) 2.0)) (sin th))
     (if (<= t_1 -0.02)
       t_2
       (if (<= t_1 1e-5)
         (/ (sin ky) (/ (sin kx) (sin th)))
         (if (<= t_1 0.9999995735978008) t_2 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	} else if (t_1 <= -0.02) {
		tmp = t_2;
	} else if (t_1 <= 1e-5) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else if (t_1 <= 0.9999995735978008) {
		tmp = t_2;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_2 = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (Math.sin(ky) / (Math.sqrt(((1.0 - Math.cos((2.0 * ky))) * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_1 <= -0.02) {
		tmp = t_2;
	} else if (t_1 <= 1e-5) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else if (t_1 <= 0.9999995735978008) {
		tmp = t_2;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_2 = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	tmp = 0
	if t_1 <= -0.999:
		tmp = (math.sin(ky) / (math.sqrt(((1.0 - math.cos((2.0 * ky))) * 2.0)) / 2.0)) * math.sin(th)
	elif t_1 <= -0.02:
		tmp = t_2
	elif t_1 <= 1e-5:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	elif t_1 <= 0.9999995735978008:
		tmp = t_2
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 2.0)) / 2.0)) * sin(th));
	elseif (t_1 <= -0.02)
		tmp = t_2;
	elseif (t_1 <= 1e-5)
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	elseif (t_1 <= 0.9999995735978008)
		tmp = t_2;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_2 = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	tmp = 0.0;
	if (t_1 <= -0.999)
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	elseif (t_1 <= -0.02)
		tmp = t_2;
	elseif (t_1 <= 1e-5)
		tmp = sin(ky) / (sin(kx) / sin(th));
	elseif (t_1 <= 0.9999995735978008)
		tmp = t_2;
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$2, If[LessEqual[t$95$1, 1e-5], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999995735978008], t$95$2, N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 86.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004 or 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999999573597800784

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.3

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 55.2

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

   9. Applied rewrites 55.2%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000008e-5

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.5

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.6

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

      8. lift-hypot.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-hypot.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 63.8

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

   9. Applied rewrites 63.8%

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

   if 0.999999573597800784 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.0%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 95.1

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

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999995735978008:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := 1 - \cos \left(2 \cdot ky\right)\\ t_3 := \left(\left(th \cdot \sin ky\right) \cdot 2\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - t\_2\right)}}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{t\_2 \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;t\_1 \leq 0.9999995735978008:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (- 1.0 (cos (* 2.0 ky))))
        (t_3
         (*
          (* (* th (sin ky)) 2.0)
          (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) t_2)))))))
   (if (<= t_1 -0.999)
     (* (/ (sin ky) (/ (sqrt (* t_2 2.0)) 2.0)) (sin th))
     (if (<= t_1 -0.02)
       t_3
       (if (<= t_1 1e-5)
         (/ (sin ky) (/ (sin kx) (sin th)))
         (if (<= t_1 0.9999995735978008) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = 1.0 - cos((2.0 * ky));
	double t_3 = ((th * sin(ky)) * 2.0) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - t_2))));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(ky) / (sqrt((t_2 * 2.0)) / 2.0)) * sin(th);
	} else if (t_1 <= -0.02) {
		tmp = t_3;
	} else if (t_1 <= 1e-5) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else if (t_1 <= 0.9999995735978008) {
		tmp = t_3;
	} 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    t_2 = 1.0d0 - cos((2.0d0 * ky))
    t_3 = ((th * sin(ky)) * 2.0d0) * sqrt((0.5d0 / (1.0d0 - (cos((2.0d0 * kx)) - t_2))))
    if (t_1 <= (-0.999d0)) then
        tmp = (sin(ky) / (sqrt((t_2 * 2.0d0)) / 2.0d0)) * sin(th)
    else if (t_1 <= (-0.02d0)) then
        tmp = t_3
    else if (t_1 <= 1d-5) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else if (t_1 <= 0.9999995735978008d0) then
        tmp = t_3
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_2 = 1.0 - Math.cos((2.0 * ky));
	double t_3 = ((th * Math.sin(ky)) * 2.0) * Math.sqrt((0.5 / (1.0 - (Math.cos((2.0 * kx)) - t_2))));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (Math.sin(ky) / (Math.sqrt((t_2 * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_1 <= -0.02) {
		tmp = t_3;
	} else if (t_1 <= 1e-5) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else if (t_1 <= 0.9999995735978008) {
		tmp = t_3;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_2 = 1.0 - math.cos((2.0 * ky))
	t_3 = ((th * math.sin(ky)) * 2.0) * math.sqrt((0.5 / (1.0 - (math.cos((2.0 * kx)) - t_2))))
	tmp = 0
	if t_1 <= -0.999:
		tmp = (math.sin(ky) / (math.sqrt((t_2 * 2.0)) / 2.0)) * math.sin(th)
	elif t_1 <= -0.02:
		tmp = t_3
	elif t_1 <= 1e-5:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	elif t_1 <= 0.9999995735978008:
		tmp = t_3
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = Float64(1.0 - cos(Float64(2.0 * ky)))
	t_3 = Float64(Float64(Float64(th * sin(ky)) * 2.0) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - t_2)))))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(t_2 * 2.0)) / 2.0)) * sin(th));
	elseif (t_1 <= -0.02)
		tmp = t_3;
	elseif (t_1 <= 1e-5)
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	elseif (t_1 <= 0.9999995735978008)
		tmp = t_3;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_2 = 1.0 - cos((2.0 * ky));
	t_3 = ((th * sin(ky)) * 2.0) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - t_2))));
	tmp = 0.0;
	if (t_1 <= -0.999)
		tmp = (sin(ky) / (sqrt((t_2 * 2.0)) / 2.0)) * sin(th);
	elseif (t_1 <= -0.02)
		tmp = t_3;
	elseif (t_1 <= 1e-5)
		tmp = sin(ky) / (sin(kx) / sin(th));
	elseif (t_1 <= 0.9999995735978008)
		tmp = t_3;
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(t$95$2 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$3, If[LessEqual[t$95$1, 1e-5], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999995735978008], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 86.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004 or 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999999573597800784

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   7. Applied rewrites 55.1%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000008e-5

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.5

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. clear-num N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      7. lower-/.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      8. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{\sin th}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin th}} \]

      10. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin th}} \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th}}} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sin kx}}{\sin th}} \]

      3. lower-sin.f64 63.8

         \[\leadsto \frac{\sin ky}{\frac{\sin kx}{\color{blue}{\sin th}}} \]

   9. Applied rewrites 63.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

   if 0.999999573597800784 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 95.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\left(\left(th \cdot \sin ky\right) \cdot 2\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999995735978008:\\ \;\;\;\;\left(\left(th \cdot \sin ky\right) \cdot 2\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ t_2 := \cos \left(2 \cdot ky\right)\\ t_3 := \sqrt{\frac{0.5}{\left(2 - t\_2\right) - \cos \left(2 \cdot kx\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - t\_2\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;t\_1 \leq 0.9999995735978008:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))
        (t_2 (cos (* 2.0 ky)))
        (t_3
         (*
          (sqrt (/ 0.5 (- (- 2.0 t_2) (cos (* 2.0 kx)))))
          (* (* th (sin ky)) 2.0))))
   (if (<= t_1 -0.999)
     (* (/ (sin ky) (/ (sqrt (* (- 1.0 t_2) 2.0)) 2.0)) (sin th))
     (if (<= t_1 -0.02)
       t_3
       (if (<= t_1 1e-5)
         (/ (sin ky) (/ (sin kx) (sin th)))
         (if (<= t_1 0.9999995735978008) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double t_2 = cos((2.0 * ky));
	double t_3 = sqrt((0.5 / ((2.0 - t_2) - cos((2.0 * kx))))) * ((th * sin(ky)) * 2.0);
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(ky) / (sqrt(((1.0 - t_2) * 2.0)) / 2.0)) * sin(th);
	} else if (t_1 <= -0.02) {
		tmp = t_3;
	} else if (t_1 <= 1e-5) {
		tmp = sin(ky) / (sin(kx) / sin(th));
	} else if (t_1 <= 0.9999995735978008) {
		tmp = t_3;
	} 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    t_2 = cos((2.0d0 * ky))
    t_3 = sqrt((0.5d0 / ((2.0d0 - t_2) - cos((2.0d0 * kx))))) * ((th * sin(ky)) * 2.0d0)
    if (t_1 <= (-0.999d0)) then
        tmp = (sin(ky) / (sqrt(((1.0d0 - t_2) * 2.0d0)) / 2.0d0)) * sin(th)
    else if (t_1 <= (-0.02d0)) then
        tmp = t_3
    else if (t_1 <= 1d-5) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else if (t_1 <= 0.9999995735978008d0) then
        tmp = t_3
    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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double t_2 = Math.cos((2.0 * ky));
	double t_3 = Math.sqrt((0.5 / ((2.0 - t_2) - Math.cos((2.0 * kx))))) * ((th * Math.sin(ky)) * 2.0);
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (Math.sin(ky) / (Math.sqrt(((1.0 - t_2) * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_1 <= -0.02) {
		tmp = t_3;
	} else if (t_1 <= 1e-5) {
		tmp = Math.sin(ky) / (Math.sin(kx) / Math.sin(th));
	} else if (t_1 <= 0.9999995735978008) {
		tmp = t_3;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	t_2 = math.cos((2.0 * ky))
	t_3 = math.sqrt((0.5 / ((2.0 - t_2) - math.cos((2.0 * kx))))) * ((th * math.sin(ky)) * 2.0)
	tmp = 0
	if t_1 <= -0.999:
		tmp = (math.sin(ky) / (math.sqrt(((1.0 - t_2) * 2.0)) / 2.0)) * math.sin(th)
	elif t_1 <= -0.02:
		tmp = t_3
	elif t_1 <= 1e-5:
		tmp = math.sin(ky) / (math.sin(kx) / math.sin(th))
	elif t_1 <= 0.9999995735978008:
		tmp = t_3
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	t_2 = cos(Float64(2.0 * ky))
	t_3 = Float64(sqrt(Float64(0.5 / Float64(Float64(2.0 - t_2) - cos(Float64(2.0 * kx))))) * Float64(Float64(th * sin(ky)) * 2.0))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(1.0 - t_2) * 2.0)) / 2.0)) * sin(th));
	elseif (t_1 <= -0.02)
		tmp = t_3;
	elseif (t_1 <= 1e-5)
		tmp = Float64(sin(ky) / Float64(sin(kx) / sin(th)));
	elseif (t_1 <= 0.9999995735978008)
		tmp = t_3;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	t_2 = cos((2.0 * ky));
	t_3 = sqrt((0.5 / ((2.0 - t_2) - cos((2.0 * kx))))) * ((th * sin(ky)) * 2.0);
	tmp = 0.0;
	if (t_1 <= -0.999)
		tmp = (sin(ky) / (sqrt(((1.0 - t_2) * 2.0)) / 2.0)) * sin(th);
	elseif (t_1 <= -0.02)
		tmp = t_3;
	elseif (t_1 <= 1e-5)
		tmp = sin(ky) / (sin(kx) / sin(th));
	elseif (t_1 <= 0.9999995735978008)
		tmp = t_3;
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(0.5 / N[(N[(2.0 - t$95$2), $MachinePrecision] - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$3, If[LessEqual[t$95$1, 1e-5], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999995735978008], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999

   1. Initial program 86.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 67.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right)} \cdot 2}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(2 \cdot ky\right)}\right) \cdot 2}}{2}} \cdot \sin th \]

      5. lower-*.f64 67.9

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(2 \cdot ky\right)}\right) \cdot 2}}{2}} \cdot \sin th \]

   7. Applied rewrites 67.9%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}}{2}} \cdot \sin th \]

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004 or 1.00000000000000008e-5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999999573597800784

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{2 \cdot \left(\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right)} \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin ky \cdot th\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin ky \cdot th\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\sin ky} \cdot th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      8. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      9. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      12. associate-+l- N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      13. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      14. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \color{blue}{\left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   7. Applied rewrites 55.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   8. Taylor expanded in kx around inf

      \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{2 - \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.1%

       \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{\left(2 - \cos \left(2 \cdot ky\right)\right) - \cos \left(2 \cdot kx\right)}} \]

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000008e-5

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.5

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. clear-num N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      7. lower-/.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      8. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{\sin th}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin th}} \]

      10. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin th}} \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th}}} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sin kx}}{\sin th}} \]

      3. lower-sin.f64 63.8

         \[\leadsto \frac{\sin ky}{\frac{\sin kx}{\color{blue}{\sin th}}} \]

   9. Applied rewrites 63.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

   if 0.999999573597800784 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 88.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 95.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\sqrt{\frac{0.5}{\left(2 - \cos \left(2 \cdot ky\right)\right) - \cos \left(2 \cdot kx\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9999995735978008:\\ \;\;\;\;\sqrt{\frac{0.5}{\left(2 - \cos \left(2 \cdot ky\right)\right) - \cos \left(2 \cdot kx\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.9999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + kx \cdot kx}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-244}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.9999)
     (*
      (/ (sin ky) (sqrt (+ (- 0.5 (* 0.5 (cos (* 2.0 ky)))) (* kx kx))))
      (sin th))
     (if (<= t_1 5e-244)
       (*
        (/ (sin ky) (/ (sqrt (* (- 1.0 (cos (* 2.0 kx))) 2.0)) 2.0))
        (sin th))
       (if (<= t_1 0.05) (/ (sin ky) (/ (sin kx) (sin th))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.9999) {
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * ky)))) + (kx * kx)))) * sin(th);
	} else if (t_1 <= 5e-244) {
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * kx))) * 2.0)) / 2.0)) * sin(th);
	} else if (t_1 <= 0.05) {
		tmp = sin(ky) / (sin(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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.9999d0)) then
        tmp = (sin(ky) / sqrt(((0.5d0 - (0.5d0 * cos((2.0d0 * ky)))) + (kx * kx)))) * sin(th)
    else if (t_1 <= 5d-244) then
        tmp = (sin(ky) / (sqrt(((1.0d0 - cos((2.0d0 * kx))) * 2.0d0)) / 2.0d0)) * sin(th)
    else if (t_1 <= 0.05d0) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.9999) {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (0.5 * Math.cos((2.0 * ky)))) + (kx * kx)))) * Math.sin(th);
	} else if (t_1 <= 5e-244) {
		tmp = (Math.sin(ky) / (Math.sqrt(((1.0 - Math.cos((2.0 * kx))) * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_1 <= 0.05) {
		tmp = Math.sin(ky) / (Math.sin(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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.9999:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (0.5 * math.cos((2.0 * ky)))) + (kx * kx)))) * math.sin(th)
	elif t_1 <= 5e-244:
		tmp = (math.sin(ky) / (math.sqrt(((1.0 - math.cos((2.0 * kx))) * 2.0)) / 2.0)) * math.sin(th)
	elif t_1 <= 0.05:
		tmp = math.sin(ky) / (math.sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.9999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky)))) + Float64(kx * kx)))) * sin(th));
	elseif (t_1 <= 5e-244)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * 2.0)) / 2.0)) * sin(th));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(ky) / Float64(sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.9999)
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * ky)))) + (kx * kx)))) * sin(th);
	elseif (t_1 <= 5e-244)
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * kx))) * 2.0)) / 2.0)) * sin(th);
	elseif (t_1 <= 0.05)
		tmp = sin(ky) / (sin(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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-244], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99990000000000001

   1. Initial program 86.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 86.3

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 86.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky \cdot 2\right)}\right)}} \cdot \sin th \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky \cdot 2\right)}\right)}} \cdot \sin th \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(ky \cdot 2\right)}\right)}} \cdot \sin th \]

      10. lower-*.f64 68.2

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \color{blue}{0.5 \cdot \cos \left(ky \cdot 2\right)}\right)}} \cdot \sin th \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky \cdot 2\right)}\right)}} \cdot \sin th \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right)}} \cdot \sin th \]

      13. lower-*.f64 68.2

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right)}} \cdot \sin th \]

   7. Applied rewrites 68.2%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   if -0.99990000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.99999999999999998e-244

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 80.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 4 \cdot {ky}^{2}}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + 4 \cdot {ky}^{2}}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, 4 \cdot {ky}^{2}\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, 4 \cdot {ky}^{2}\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, 4 \cdot {ky}^{2}\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, 4 \cdot {ky}^{2}\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{{ky}^{2} \cdot 4}\right)}}{2}} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{{ky}^{2} \cdot 4}\right)}}{2}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(ky \cdot ky\right)} \cdot 4\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 44.9

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(ky \cdot ky\right)} \cdot 4\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 44.9%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(ky \cdot ky\right) \cdot 4\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right)}}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 49.7%

       \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot \color{blue}{2}}}{2}} \cdot \sin th \]

   if 4.99999999999999998e-244 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.3

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.5

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. clear-num N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      7. lower-/.f64 99.4

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      8. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{\sin th}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin th}} \]

      10. lower-hypot.f64 99.4

          \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin th}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th}}} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sin kx}}{\sin th}} \]

      3. lower-sin.f64 42.6

         \[\leadsto \frac{\sin ky}{\frac{\sin kx}{\color{blue}{\sin th}}} \]

   9. Applied rewrites 42.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 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 70.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.9999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + kx \cdot kx}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-244}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.1)
     (* (/ (sin ky) (/ (sqrt (* (- 1.0 (cos (* 2.0 ky))) 2.0)) 2.0)) (sin th))
     (if (<= t_1 0.05) (/ (sin ky) (/ (sin kx) (sin th))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	} else if (t_1 <= 0.05) {
		tmp = sin(ky) / (sin(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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.1d0)) then
        tmp = (sin(ky) / (sqrt(((1.0d0 - cos((2.0d0 * ky))) * 2.0d0)) / 2.0d0)) * sin(th)
    else if (t_1 <= 0.05d0) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (Math.sin(ky) / (Math.sqrt(((1.0 - Math.cos((2.0 * ky))) * 2.0)) / 2.0)) * Math.sin(th);
	} else if (t_1 <= 0.05) {
		tmp = Math.sin(ky) / (Math.sin(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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.1:
		tmp = (math.sin(ky) / (math.sqrt(((1.0 - math.cos((2.0 * ky))) * 2.0)) / 2.0)) * math.sin(th)
	elif t_1 <= 0.05:
		tmp = math.sin(ky) / (math.sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 2.0)) / 2.0)) * sin(th));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(ky) / Float64(sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	elseif (t_1 <= 0.05)
		tmp = sin(ky) / (sin(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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 90.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 78.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right)} \cdot 2}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(2 \cdot ky\right)}\right) \cdot 2}}{2}} \cdot \sin th \]

      5. lower-*.f64 52.8

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(2 \cdot ky\right)}\right) \cdot 2}}{2}} \cdot \sin th \]

   7. Applied rewrites 52.8%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}}{2}} \cdot \sin th \]

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.5

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. clear-num N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      7. lower-/.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      8. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{\sin th}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin th}} \]

      10. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin th}} \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th}}} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sin kx}}{\sin th}} \]

      3. lower-sin.f64 60.5

         \[\leadsto \frac{\sin ky}{\frac{\sin kx}{\color{blue}{\sin th}}} \]

   9. Applied rewrites 60.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 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 70.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.9999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + kx \cdot kx}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.9999)
     (*
      (/ (sin ky) (sqrt (+ (- 0.5 (* 0.5 (cos (* 2.0 ky)))) (* kx kx))))
      (sin th))
     (if (<= t_1 0.05) (/ (sin ky) (/ (sin kx) (sin th))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.9999) {
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * ky)))) + (kx * kx)))) * sin(th);
	} else if (t_1 <= 0.05) {
		tmp = sin(ky) / (sin(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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.9999d0)) then
        tmp = (sin(ky) / sqrt(((0.5d0 - (0.5d0 * cos((2.0d0 * ky)))) + (kx * kx)))) * sin(th)
    else if (t_1 <= 0.05d0) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.9999) {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (0.5 * Math.cos((2.0 * ky)))) + (kx * kx)))) * Math.sin(th);
	} else if (t_1 <= 0.05) {
		tmp = Math.sin(ky) / (Math.sin(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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.9999:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (0.5 * math.cos((2.0 * ky)))) + (kx * kx)))) * math.sin(th)
	elif t_1 <= 0.05:
		tmp = math.sin(ky) / (math.sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.9999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky)))) + Float64(kx * kx)))) * sin(th));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(ky) / Float64(sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.9999)
		tmp = (sin(ky) / sqrt(((0.5 - (0.5 * cos((2.0 * ky)))) + (kx * kx)))) * sin(th);
	elseif (t_1 <= 0.05)
		tmp = sin(ky) / (sin(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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99990000000000001

   1. Initial program 86.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 86.3

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 86.3%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky} \cdot \sin ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky \cdot 2\right)}\right)}} \cdot \sin th \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky \cdot 2\right)}\right)}} \cdot \sin th \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(ky \cdot 2\right)}\right)}} \cdot \sin th \]

      10. lower-*.f64 68.2

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - \color{blue}{0.5 \cdot \cos \left(ky \cdot 2\right)}\right)}} \cdot \sin th \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky \cdot 2\right)}\right)}} \cdot \sin th \]

      12. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right)}} \cdot \sin th \]

      13. lower-*.f64 68.2

          \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot ky\right)}\right)}} \cdot \sin th \]

   7. Applied rewrites 68.2%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   if -0.99990000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.4

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.5

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. clear-num N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      7. lower-/.f64 99.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      8. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{\sin th}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin th}} \]

      10. lower-hypot.f64 99.5

          \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin th}} \]

   6. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th}}} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sin kx}}{\sin th}} \]

      3. lower-sin.f64 48.6

         \[\leadsto \frac{\sin ky}{\frac{\sin kx}{\color{blue}{\sin th}}} \]

   9. Applied rewrites 48.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 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 70.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.9999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + kx \cdot kx}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.52:\\ \;\;\;\;\sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.52)
     (* (sqrt (/ 0.5 (- 1.0 (cos (* 2.0 ky))))) (* (* th (sin ky)) 2.0))
     (if (<= t_1 0.05) (/ (sin ky) (/ (sin kx) (sin th))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.52) {
		tmp = sqrt((0.5 / (1.0 - cos((2.0 * ky))))) * ((th * sin(ky)) * 2.0);
	} else if (t_1 <= 0.05) {
		tmp = sin(ky) / (sin(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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.52d0)) then
        tmp = sqrt((0.5d0 / (1.0d0 - cos((2.0d0 * ky))))) * ((th * sin(ky)) * 2.0d0)
    else if (t_1 <= 0.05d0) then
        tmp = sin(ky) / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.52) {
		tmp = Math.sqrt((0.5 / (1.0 - Math.cos((2.0 * ky))))) * ((th * Math.sin(ky)) * 2.0);
	} else if (t_1 <= 0.05) {
		tmp = Math.sin(ky) / (Math.sin(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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.52:
		tmp = math.sqrt((0.5 / (1.0 - math.cos((2.0 * ky))))) * ((th * math.sin(ky)) * 2.0)
	elif t_1 <= 0.05:
		tmp = math.sin(ky) / (math.sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.52)
		tmp = Float64(sqrt(Float64(0.5 / Float64(1.0 - cos(Float64(2.0 * ky))))) * Float64(Float64(th * sin(ky)) * 2.0));
	elseif (t_1 <= 0.05)
		tmp = Float64(sin(ky) / Float64(sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.52)
		tmp = sqrt((0.5 / (1.0 - cos((2.0 * ky))))) * ((th * sin(ky)) * 2.0);
	elseif (t_1 <= 0.05)
		tmp = sin(ky) / (sin(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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.52], N[(N[Sqrt[N[(0.5 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.52000000000000002

   1. Initial program 90.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 76.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{2 \cdot \left(\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right)} \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin ky \cdot th\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin ky \cdot th\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\sin ky} \cdot th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      8. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      9. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      12. associate-+l- N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      13. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      14. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \color{blue}{\left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   7. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \cos \left(2 \cdot ky\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 30.2%

       \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \]

   if -0.52000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.5

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. clear-num N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      7. lower-/.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      8. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{\sin th}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin th}} \]

      10. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin th}} \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th}}} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sin kx}}{\sin th}} \]

      3. lower-sin.f64 57.6

         \[\leadsto \frac{\sin ky}{\frac{\sin kx}{\color{blue}{\sin th}}} \]

   9. Applied rewrites 57.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sin kx}{\sin th}}} \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 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 70.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.52:\\ \;\;\;\;\sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.52:\\ \;\;\;\;\sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.52)
     (* (sqrt (/ 0.5 (- 1.0 (cos (* 2.0 ky))))) (* (* th (sin ky)) 2.0))
     (if (<= t_1 0.05) (* (/ (sin ky) (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.52) {
		tmp = sqrt((0.5 / (1.0 - cos((2.0 * ky))))) * ((th * sin(ky)) * 2.0);
	} else if (t_1 <= 0.05) {
		tmp = (sin(ky) / sin(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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.52d0)) then
        tmp = sqrt((0.5d0 / (1.0d0 - cos((2.0d0 * ky))))) * ((th * sin(ky)) * 2.0d0)
    else if (t_1 <= 0.05d0) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.52) {
		tmp = Math.sqrt((0.5 / (1.0 - Math.cos((2.0 * ky))))) * ((th * Math.sin(ky)) * 2.0);
	} else if (t_1 <= 0.05) {
		tmp = (Math.sin(ky) / Math.sin(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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.52:
		tmp = math.sqrt((0.5 / (1.0 - math.cos((2.0 * ky))))) * ((th * math.sin(ky)) * 2.0)
	elif t_1 <= 0.05:
		tmp = (math.sin(ky) / math.sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.52)
		tmp = Float64(sqrt(Float64(0.5 / Float64(1.0 - cos(Float64(2.0 * ky))))) * Float64(Float64(th * sin(ky)) * 2.0));
	elseif (t_1 <= 0.05)
		tmp = Float64(Float64(sin(ky) / sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.52)
		tmp = sqrt((0.5 / (1.0 - cos((2.0 * ky))))) * ((th * sin(ky)) * 2.0);
	elseif (t_1 <= 0.05)
		tmp = (sin(ky) / sin(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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.52], N[(N[Sqrt[N[(0.5 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.52000000000000002

   1. Initial program 90.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 76.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{2 \cdot \left(\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right)} \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin ky \cdot th\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin ky \cdot th\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\sin ky} \cdot th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      8. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      9. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      12. associate-+l- N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      13. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      14. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \color{blue}{\left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   7. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \cos \left(2 \cdot ky\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 30.2%

       \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \]

   if -0.52000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 57.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 57.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 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 70.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.52:\\ \;\;\;\;\sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.52:\\ \;\;\;\;\sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.52)
     (* (sqrt (/ 0.5 (- 1.0 (cos (* 2.0 ky))))) (* (* th (sin ky)) 2.0))
     (if (<= t_1 0.05) (* (/ (sin th) (sin kx)) (sin ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.52) {
		tmp = sqrt((0.5 / (1.0 - cos((2.0 * ky))))) * ((th * sin(ky)) * 2.0);
	} else if (t_1 <= 0.05) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} 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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.52d0)) then
        tmp = sqrt((0.5d0 / (1.0d0 - cos((2.0d0 * ky))))) * ((th * sin(ky)) * 2.0d0)
    else if (t_1 <= 0.05d0) then
        tmp = (sin(th) / sin(kx)) * sin(ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.52) {
		tmp = Math.sqrt((0.5 / (1.0 - Math.cos((2.0 * ky))))) * ((th * Math.sin(ky)) * 2.0);
	} else if (t_1 <= 0.05) {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.52:
		tmp = math.sqrt((0.5 / (1.0 - math.cos((2.0 * ky))))) * ((th * math.sin(ky)) * 2.0)
	elif t_1 <= 0.05:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.52)
		tmp = Float64(sqrt(Float64(0.5 / Float64(1.0 - cos(Float64(2.0 * ky))))) * Float64(Float64(th * sin(ky)) * 2.0));
	elseif (t_1 <= 0.05)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.52)
		tmp = sqrt((0.5 / (1.0 - cos((2.0 * ky))))) * ((th * sin(ky)) * 2.0);
	elseif (t_1 <= 0.05)
		tmp = (sin(th) / sin(kx)) * sin(ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.52], N[(N[Sqrt[N[(0.5 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.52000000000000002

   1. Initial program 90.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 76.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{2 \cdot \left(\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right)} \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin ky \cdot th\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin ky \cdot th\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\sin ky} \cdot th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      8. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      9. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      12. associate-+l- N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      13. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      14. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \color{blue}{\left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   7. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \cos \left(2 \cdot ky\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 30.2%

       \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \]

   if -0.52000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.5

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.6

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th}}{\sin kx} \cdot \sin ky \]

      3. lower-sin.f64 57.6

         \[\leadsto \frac{\sin th}{\color{blue}{\sin kx}} \cdot \sin ky \]

   7. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\sin kx}} \cdot \sin ky \]

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 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 70.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.52:\\ \;\;\;\;\sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.02)
     (* (sqrt (/ 0.5 (- 1.0 (cos (* 2.0 ky))))) (* (* th (sin ky)) 2.0))
     (if (<= t_1 0.005) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = sqrt((0.5 / (1.0 - cos((2.0 * ky))))) * ((th * sin(ky)) * 2.0);
	} else if (t_1 <= 0.005) {
		tmp = (ky / sin(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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.02d0)) then
        tmp = sqrt((0.5d0 / (1.0d0 - cos((2.0d0 * ky))))) * ((th * sin(ky)) * 2.0d0)
    else if (t_1 <= 0.005d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = Math.sqrt((0.5 / (1.0 - Math.cos((2.0 * ky))))) * ((th * Math.sin(ky)) * 2.0);
	} else if (t_1 <= 0.005) {
		tmp = (ky / Math.sin(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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.02:
		tmp = math.sqrt((0.5 / (1.0 - math.cos((2.0 * ky))))) * ((th * math.sin(ky)) * 2.0)
	elif t_1 <= 0.005:
		tmp = (ky / math.sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(sqrt(Float64(0.5 / Float64(1.0 - cos(Float64(2.0 * ky))))) * Float64(Float64(th * sin(ky)) * 2.0));
	elseif (t_1 <= 0.005)
		tmp = Float64(Float64(ky / sin(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(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.02)
		tmp = sqrt((0.5 / (1.0 - cos((2.0 * ky))))) * ((th * sin(ky)) * 2.0);
	elseif (t_1 <= 0.005)
		tmp = (ky / sin(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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.02], N[(N[Sqrt[N[(0.5 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.005], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 90.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 79.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{2 \cdot \left(\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(th \cdot \sin ky\right)\right)} \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin ky \cdot th\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin ky \cdot th\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\sin ky} \cdot th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      8. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      9. associate-/r* N/A

         \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      10. metadata-eval N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      12. associate-+l- N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      13. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      14. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \color{blue}{\left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   7. Applied rewrites 44.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \cos \left(2 \cdot ky\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.2%

       \[\leadsto \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \]

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 63.7

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0050000000000000001 < (/.f64 (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 92.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 69.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.02:\\ \;\;\;\;\sqrt{\frac{0.5}{1 - \cos \left(2 \cdot ky\right)}} \cdot \left(\left(th \cdot \sin ky\right) \cdot 2\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.005:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.005:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.005)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.005) {
		tmp = (ky / sin(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) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.005d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.005) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.005:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.005)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.005)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 33.0

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 33.0%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0050000000000000001 < (/.f64 (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 92.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 69.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.005:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\frac{\sqrt{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) - \left(\cos \left(2 \cdot ky\right) - 1\right)\right) \cdot 2}}{2}}{\sin th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2e-15)
   (*
    (/
     (sin ky)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin th))
   (/
    (sin ky)
    (/
     (/
      (sqrt (* (- (- 1.0 (cos (* 2.0 kx))) (- (cos (* 2.0 ky)) 1.0)) 2.0))
      2.0)
     (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 2e-15) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = sin(ky) / ((sqrt((((1.0 - cos((2.0 * kx))) - (cos((2.0 * ky)) - 1.0)) * 2.0)) / 2.0) / sin(th));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 2e-15)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(sin(ky) / Float64(Float64(sqrt(Float64(Float64(Float64(1.0 - cos(Float64(2.0 * kx))) - Float64(cos(Float64(2.0 * ky)) - 1.0)) * 2.0)) / 2.0) / sin(th)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 2e-15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[Sqrt[N[(N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.0000000000000002e-15

   1. Initial program 89.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.9

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if 2.0000000000000002e-15 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

      7. lower-/.f64 99.4

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      10. +-commutative N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

      11. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

      13. lift-pow.f64 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

      15. lower-hypot.f64 99.4

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

      4. clear-num N/A

         \[\leadsto \sin ky \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      7. lower-/.f64 99.4

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin th}}} \]

      8. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{\sin th}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{\sin th}} \]

      10. lower-hypot.f64 99.4

          \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin th}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th}}} \]

   8. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\frac{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}{2}}}{\sin th}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\frac{\sqrt{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) - \left(\cos \left(2 \cdot ky\right) - 1\right)\right) \cdot 2}}{2}}{\sin th}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.005:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.005)
   (* (* (fma (* th th) -0.16666666666666666 1.0) th) (/ ky (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.005) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.005)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0050000000000000001

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. 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 54.2

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th\right) \]

   7. Applied rewrites 54.2%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

      2. lower-sin.f64 20.7

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   10. Applied rewrites 20.7%

       \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   if 0.0050000000000000001 < (/.f64 (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 92.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 69.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.005:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \sin ky}{\sqrt{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) - \left(\cos \left(2 \cdot ky\right) - 1\right)\right) \cdot 2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2e-15)
   (*
    (/
     (sin ky)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin th))
   (*
    (/
     (* 2.0 (sin ky))
     (sqrt (* (- (- 1.0 (cos (* 2.0 kx))) (- (cos (* 2.0 ky)) 1.0)) 2.0)))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 2e-15) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = ((2.0 * sin(ky)) / sqrt((((1.0 - cos((2.0 * kx))) - (cos((2.0 * ky)) - 1.0)) * 2.0))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 2e-15)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(Float64(2.0 * sin(ky)) / sqrt(Float64(Float64(Float64(1.0 - cos(Float64(2.0 * kx))) - Float64(cos(Float64(2.0 * ky)) - 1.0)) * 2.0))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 2e-15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.0000000000000002e-15

   1. Initial program 89.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.9

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if 2.0000000000000002e-15 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot 2}{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \sin ky}{\sqrt{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) - \left(\cos \left(2 \cdot ky\right) - 1\right)\right) \cdot 2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}} \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2e-15)
   (*
    (/
     (sin ky)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin th))
   (*
    (*
     (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) (- 1.0 (cos (* 2.0 ky)))))))
     (* 2.0 (sin ky)))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 2e-15) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = (sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky))))))) * (2.0 * sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 2e-15)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - Float64(1.0 - cos(Float64(2.0 * ky))))))) * Float64(2.0 * sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 2e-15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.0000000000000002e-15

   1. Initial program 89.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.9

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if 2.0000000000000002e-15 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)} \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \sin ky\right)} \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right) \cdot \sin th \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\sin ky}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right) \cdot \sin th \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \sin ky\right) \cdot \color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}\right) \cdot \sin th \]

      6. distribute-lft-out N/A

         \[\leadsto \left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}}\right) \cdot \sin th \]

      7. associate-/r* N/A

         \[\leadsto \left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}}\right) \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right) \cdot \sin th \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}}\right) \cdot \sin th \]

      10. associate-+l- N/A

          \[\leadsto \left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}}\right) \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}}\right) \cdot \sin th \]

      12. lower--.f64 N/A

          \[\leadsto \left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \color{blue}{\left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}}\right) \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \left(\color{blue}{\cos \left(2 \cdot kx\right)} - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\right) \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\frac{\frac{1}{2}}{1 - \left(\cos \color{blue}{\left(2 \cdot kx\right)} - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\right) \cdot \sin th \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin ky\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\right)} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}} \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin th \cdot \sin ky\right) \cdot 2\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2e-15)
   (*
    (/
     (sin ky)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin th))
   (*
    (* (* (sin th) (sin ky)) 2.0)
    (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) (- 1.0 (cos (* 2.0 ky))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 2e-15) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = ((sin(th) * sin(ky)) * 2.0) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 2e-15)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(Float64(sin(th) * sin(ky)) * 2.0) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - Float64(1.0 - cos(Float64(2.0 * ky))))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 2e-15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 2.0000000000000002e-15

   1. Initial program 89.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.9

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if 2.0000000000000002e-15 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{2 \cdot \left(\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot \sin th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot \sin th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot \sin th\right)\right)} \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      9. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      10. associate-/r* N/A

          \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      11. metadata-eval N/A

          \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      13. associate-+l- N/A

          \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      14. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin th \cdot \sin ky\right) \cdot 2\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 15.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sin th \leq 5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;1 \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))
       (sin th))
      5e-310)
   (* (* (* -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(ky), 2.0) + pow(sin(kx), 2.0)))) * sin(th)) <= 5e-310) {
		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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) * sin(th)) <= 5d-310) 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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) * Math.sin(th)) <= 5e-310) {
		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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) * math.sin(th)) <= 5e-310:
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) * sin(th)) <= 5e-310)
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) * sin(th)) <= 5e-310)
		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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 5e-310], 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)) < 4.999999999999985e-310

   1. Initial program 92.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 24.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 13.8%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   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 20.0%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 20.0%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 4.999999999999985e-310 < (*.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 95.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 28.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\sin th} \]

   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 15.5%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   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 4.2%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   12. Taylor expanded in th around 0

       \[\leadsto 1 \cdot th \]
   13. Step-by-step derivation

   14. Applied rewrites 15.4%

       \[\leadsto 1 \cdot th \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sin th \leq 5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;1 \cdot th\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 30.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 3.7 \cdot 10^{-90}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 3.7e-90)
   (* (* (* -0.16666666666666666 th) th) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 3.7e-90) {
		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(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 3.7d-90) 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(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 3.7e-90) {
		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(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 3.7e-90:
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 3.7e-90)
		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(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 3.7e-90)
		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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.7e-90], 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))))) < 3.70000000000000018e-90

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   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 17.6%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 17.6%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th \]

   if 3.70000000000000018e-90 < (/.f64 (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 92.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 3.7 \cdot 10^{-90}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot th\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 13.5% 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 94.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 26.4

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 26.4%

   \[\leadsto \color{blue}{\sin th} \]

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 14.6%

   \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

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 12.2%

    \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

12. Taylor expanded in th around 0

    \[\leadsto 1 \cdot th \]
13. Step-by-step derivation

14. Applied rewrites 14.7%

    \[\leadsto 1 \cdot th \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024297 
(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)))