Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 10.0s

Alternatives: 16

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

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 92.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   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%

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

5. Final simplification 99.6%

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

Alternative 2: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_1 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2 \cdot \left(\mathsf{fma}\left(th \cdot th, 0.16666666666666666, -1\right) \cdot \left(th \cdot \sin ky\right)\right)\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.992:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{\mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}}\right)}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ -1.0 (hypot (sin ky) (sin kx))))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_1 t_3)))))
   (if (<= t_4 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_4 -0.1)
       (* t_2 (* (fma (* th th) 0.16666666666666666 -1.0) (* th (sin ky))))
       (if (<= t_4 0.05)
         (* (/ (sin ky) (sqrt (+ (* ky ky) t_3))) (sin th))
         (if (<= t_4 0.992)
           (* (* (- th) (sin ky)) t_2)
           (/
            (sin th)
            (/
             (hypot
              (sin ky)
              (/ 1.0 (/ (fma 0.16666666666666666 (* kx kx) 1.0) kx)))
             (sin ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = -1.0 / hypot(sin(ky), sin(kx));
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_1 + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2 * (fma((th * th), 0.16666666666666666, -1.0) * (th * sin(ky)));
	} else if (t_4 <= 0.05) {
		tmp = (sin(ky) / sqrt(((ky * ky) + t_3))) * sin(th);
	} else if (t_4 <= 0.992) {
		tmp = (-th * sin(ky)) * t_2;
	} else {
		tmp = sin(th) / (hypot(sin(ky), (1.0 / (fma(0.16666666666666666, (kx * kx), 1.0) / kx))) / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_1 + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = Float64(t_2 * Float64(fma(Float64(th * th), 0.16666666666666666, -1.0) * Float64(th * sin(ky))));
	elseif (t_4 <= 0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + t_3))) * sin(th));
	elseif (t_4 <= 0.992)
		tmp = Float64(Float64(Float64(-th) * sin(ky)) * t_2);
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), Float64(1.0 / Float64(fma(0.16666666666666666, Float64(kx * kx), 1.0) / kx))) / sin(ky)));
	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[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.1], N[(t$95$2 * N[(N[(N[(th * th), $MachinePrecision] * 0.16666666666666666 + -1.0), $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.992], N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(1.0 / N[(N[(0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 85.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

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

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.3

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

      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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 51.5%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 98.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99199999999999999

   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. 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 99.3

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

      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 99.2%

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

   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 54.2

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

   7. Applied rewrites 54.2%

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

   if 0.99199999999999999 < (/.f64 (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 76.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 76.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, 0.16666666666666666, -1\right) \cdot \left(th \cdot \sin ky\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.992:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{\mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}}\right)}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_1 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2 \cdot \left(\mathsf{fma}\left(th \cdot th, 0.16666666666666666, -1\right) \cdot \left(th \cdot \sin ky\right)\right)\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{1}{kx}}\right)}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ -1.0 (hypot (sin ky) (sin kx))))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_1 t_3)))))
   (if (<= t_4 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_4 -0.1)
       (* t_2 (* (fma (* th th) 0.16666666666666666 -1.0) (* th (sin ky))))
       (if (<= t_4 0.05)
         (* (/ (sin ky) (sqrt (+ (* ky ky) t_3))) (sin th))
         (if (<= t_4 0.995)
           (* (* (- th) (sin ky)) t_2)
           (/ (sin th) (/ (hypot (sin ky) (/ 1.0 (/ 1.0 kx))) (sin ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = -1.0 / hypot(sin(ky), sin(kx));
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_1 + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2 * (fma((th * th), 0.16666666666666666, -1.0) * (th * sin(ky)));
	} else if (t_4 <= 0.05) {
		tmp = (sin(ky) / sqrt(((ky * ky) + t_3))) * sin(th);
	} else if (t_4 <= 0.995) {
		tmp = (-th * sin(ky)) * t_2;
	} else {
		tmp = sin(th) / (hypot(sin(ky), (1.0 / (1.0 / kx))) / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_1 + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = Float64(t_2 * Float64(fma(Float64(th * th), 0.16666666666666666, -1.0) * Float64(th * sin(ky))));
	elseif (t_4 <= 0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + t_3))) * sin(th));
	elseif (t_4 <= 0.995)
		tmp = Float64(Float64(Float64(-th) * sin(ky)) * t_2);
	else
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), Float64(1.0 / Float64(1.0 / kx))) / sin(ky)));
	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[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.1], N[(t$95$2 * N[(N[(N[(th * th), $MachinePrecision] * 0.16666666666666666 + -1.0), $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(1.0 / N[(1.0 / kx), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 85.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

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

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.3

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

      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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 51.5%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 98.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   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. 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 99.3

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

      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 99.2%

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

   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 54.2

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

   7. Applied rewrites 54.2%

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

   if 0.994999999999999996 < (/.f64 (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 76.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 76.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in kx around 0

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

      1. lower-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, 0.16666666666666666, -1\right) \cdot \left(th \cdot \sin ky\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{1}{kx}}\right)}{\sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_1 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2 \cdot \left(\mathsf{fma}\left(th \cdot th, 0.16666666666666666, -1\right) \cdot \left(th \cdot \sin ky\right)\right)\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9998589899959065:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ -1.0 (hypot (sin ky) (sin kx))))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_1 t_3)))))
   (if (<= t_4 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_4 -0.1)
       (* t_2 (* (fma (* th th) 0.16666666666666666 -1.0) (* th (sin ky))))
       (if (<= t_4 0.05)
         (* (/ (sin ky) (sqrt (+ (* ky ky) t_3))) (sin th))
         (if (<= t_4 0.9998589899959065)
           (* (* (- th) (sin ky)) t_2)
           (*
            (/ (sin ky) (fma (* 0.5 kx) (/ kx (sin ky)) (sin ky)))
            (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = -1.0 / hypot(sin(ky), sin(kx));
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_1 + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2 * (fma((th * th), 0.16666666666666666, -1.0) * (th * sin(ky)));
	} else if (t_4 <= 0.05) {
		tmp = (sin(ky) / sqrt(((ky * ky) + t_3))) * sin(th);
	} else if (t_4 <= 0.9998589899959065) {
		tmp = (-th * sin(ky)) * t_2;
	} else {
		tmp = (sin(ky) / fma((0.5 * kx), (kx / sin(ky)), sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_1 + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = Float64(t_2 * Float64(fma(Float64(th * th), 0.16666666666666666, -1.0) * Float64(th * sin(ky))));
	elseif (t_4 <= 0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + t_3))) * sin(th));
	elseif (t_4 <= 0.9998589899959065)
		tmp = Float64(Float64(Float64(-th) * sin(ky)) * t_2);
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(0.5 * kx), Float64(kx / sin(ky)), sin(ky))) * 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[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.1], N[(t$95$2 * N[(N[(N[(th * th), $MachinePrecision] * 0.16666666666666666 + -1.0), $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9998589899959065], N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(0.5 * kx), $MachinePrecision] * N[(kx / N[Sin[ky], $MachinePrecision]), $MachinePrecision] + N[Sin[ky], $MachinePrecision]), $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))))) < -1

   1. Initial program 85.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

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

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.3

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

      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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 51.5%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 98.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.9998589899959065

   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. 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 99.3

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

      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 99.2%

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

   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 54.2

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

   7. Applied rewrites 54.2%

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

   if 0.9998589899959065 < (/.f64 (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 76.7%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, 0.16666666666666666, -1\right) \cdot \left(th \cdot \sin ky\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9998589899959065:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(0.5 \cdot kx, \frac{kx}{\sin ky}, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_1 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_2 \cdot \left(\mathsf{fma}\left(th \cdot th, 0.16666666666666666, -1\right) \cdot \left(th \cdot \sin ky\right)\right)\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9998589899959065:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ -1.0 (hypot (sin ky) (sin kx))))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_1 t_3)))))
   (if (<= t_4 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_4 -0.1)
       (* t_2 (* (fma (* th th) 0.16666666666666666 -1.0) (* th (sin ky))))
       (if (<= t_4 0.05)
         (* (/ (sin ky) (sqrt (+ (* ky ky) t_3))) (sin th))
         (if (<= t_4 0.9998589899959065)
           (* (* (- th) (sin ky)) t_2)
           (* (/ (sin ky) (fma (* (/ kx ky) 0.5) kx (sin ky))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = -1.0 / hypot(sin(ky), sin(kx));
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_1 + t_3));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_4 <= -0.1) {
		tmp = t_2 * (fma((th * th), 0.16666666666666666, -1.0) * (th * sin(ky)));
	} else if (t_4 <= 0.05) {
		tmp = (sin(ky) / sqrt(((ky * ky) + t_3))) * sin(th);
	} else if (t_4 <= 0.9998589899959065) {
		tmp = (-th * sin(ky)) * t_2;
	} else {
		tmp = (sin(ky) / fma(((kx / ky) * 0.5), kx, sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_1 + t_3)))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_4 <= -0.1)
		tmp = Float64(t_2 * Float64(fma(Float64(th * th), 0.16666666666666666, -1.0) * Float64(th * sin(ky))));
	elseif (t_4 <= 0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + t_3))) * sin(th));
	elseif (t_4 <= 0.9998589899959065)
		tmp = Float64(Float64(Float64(-th) * sin(ky)) * t_2);
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(Float64(kx / ky) * 0.5), kx, sin(ky))) * 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[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.1], N[(t$95$2 * N[(N[(N[(th * th), $MachinePrecision] * 0.16666666666666666 + -1.0), $MachinePrecision] * N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9998589899959065], N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(kx / ky), $MachinePrecision] * 0.5), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $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))))) < -1

   1. Initial program 85.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

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

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.3

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

      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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

   7. Applied rewrites 51.5%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 98.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.9998589899959065

   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. 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 99.3

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

      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 99.2%

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

   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 54.2

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

   7. Applied rewrites 54.2%

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

   if 0.9998589899959065 < (/.f64 (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 76.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{kx}{ky}, kx, \sin ky\right)} \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 88.6%

       \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, 0.16666666666666666, -1\right) \cdot \left(th \cdot \sin ky\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9998589899959065:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + t\_1}}\\ t_4 := \left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9998589899959065:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 t_1))))
        (t_4 (* (* (- th) (sin ky)) (/ -1.0 (hypot (sin ky) (sin kx))))))
   (if (<= t_3 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_2))) (sin th))
     (if (<= t_3 -0.1)
       t_4
       (if (<= t_3 0.05)
         (* (/ (sin ky) (sqrt (+ (* ky ky) t_1))) (sin th))
         (if (<= t_3 0.9998589899959065)
           t_4
           (* (/ (sin ky) (fma (* (/ kx ky) 0.5) kx (sin ky))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + t_1));
	double t_4 = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = t_4;
	} else if (t_3 <= 0.05) {
		tmp = (sin(ky) / sqrt(((ky * ky) + t_1))) * sin(th);
	} else if (t_3 <= 0.9998589899959065) {
		tmp = t_4;
	} else {
		tmp = (sin(ky) / fma(((kx / ky) * 0.5), kx, sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + t_1)))
	t_4 = Float64(Float64(Float64(-th) * sin(ky)) * Float64(-1.0 / hypot(sin(ky), sin(kx))))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_2))) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = t_4;
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + t_1))) * sin(th));
	elseif (t_3 <= 0.9998589899959065)
		tmp = t_4;
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(Float64(kx / ky) * 0.5), kx, sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], 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, -0.1], t$95$4, If[LessEqual[t$95$3, 0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9998589899959065], t$95$4, N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(kx / ky), $MachinePrecision] * 0.5), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 85.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.9998589899959065

   1. Initial program 99.3%

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

      1. lift-*.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 99.3

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

      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 52.7

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

   7. Applied rewrites 52.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 98.6%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.9998589899959065 < (/.f64 (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 76.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{kx}{ky}, kx, \sin ky\right)} \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 88.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + {\sin kx}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9998589899959065:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin kx}^{2}}}\\ t_4 := \left(\left(-th\right) \cdot \sin ky\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 0.9998589899959065:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

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)))))
        (t_4 (* (* (- th) (sin ky)) t_1)))
   (if (<= t_3 -1.0)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_2))) (sin th))
     (if (<= t_3 -0.1)
       t_4
       (if (<= t_3 0.05)
         (* (* (- ky) (sin th)) t_1)
         (if (<= t_3 0.9998589899959065)
           t_4
           (* (/ (sin ky) (fma (* (/ kx ky) 0.5) kx (sin ky))) (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 t_4 = (-th * sin(ky)) * t_1;
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = t_4;
	} else if (t_3 <= 0.05) {
		tmp = (-ky * sin(th)) * t_1;
	} else if (t_3 <= 0.9998589899959065) {
		tmp = t_4;
	} else {
		tmp = (sin(ky) / fma(((kx / ky) * 0.5), kx, sin(ky))) * 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))))
	t_4 = Float64(Float64(Float64(-th) * sin(ky)) * t_1)
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_2))) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = t_4;
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(Float64(-ky) * sin(th)) * t_1);
	elseif (t_3 <= 0.9998589899959065)
		tmp = t_4;
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(Float64(kx / ky) * 0.5), kx, sin(ky))) * 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]}, Block[{t$95$4 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], t$95$4, If[LessEqual[t$95$3, 0.05], N[(N[((-ky) * N[Sin[th], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.9998589899959065], t$95$4, N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(kx / ky), $MachinePrecision] * 0.5), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 85.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.9998589899959065

   1. Initial program 99.3%

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

      1. lift-*.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 99.3

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

      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 52.7

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

   7. Applied rewrites 52.7%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 98.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 98.0

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

      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 98.6%

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

   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 97.5

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

   7. Applied rewrites 97.5%

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

   if 0.9998589899959065 < (/.f64 (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 76.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{kx}{ky}, kx, \sin ky\right)} \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 88.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9998589899959065:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := \left(\left(-th\right) \cdot \sin ky\right) \cdot t\_1\\ t_3 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq 0.9998589899959065:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ -1.0 (hypot (sin ky) (sin kx))))
        (t_2 (* (* (- th) (sin ky)) t_1))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_3 -0.1)
     t_2
     (if (<= t_3 0.05)
       (* (* (- ky) (sin th)) t_1)
       (if (<= t_3 0.9998589899959065)
         t_2
         (* (/ (sin ky) (fma (* (/ kx ky) 0.5) kx (sin ky))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = -1.0 / hypot(sin(ky), sin(kx));
	double t_2 = (-th * sin(ky)) * t_1;
	double t_3 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_3 <= -0.1) {
		tmp = t_2;
	} else if (t_3 <= 0.05) {
		tmp = (-ky * sin(th)) * t_1;
	} else if (t_3 <= 0.9998589899959065) {
		tmp = t_2;
	} else {
		tmp = (sin(ky) / fma(((kx / ky) * 0.5), kx, sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(-1.0 / hypot(sin(ky), sin(kx)))
	t_2 = Float64(Float64(Float64(-th) * sin(ky)) * t_1)
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.1)
		tmp = t_2;
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(Float64(-ky) * sin(th)) * t_1);
	elseif (t_3 <= 0.9998589899959065)
		tmp = t_2;
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(Float64(kx / ky) * 0.5), kx, sin(ky))) * 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[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = 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$3, -0.1], t$95$2, If[LessEqual[t$95$3, 0.05], N[(N[((-ky) * N[Sin[th], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.9998589899959065], t$95$2, N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(kx / ky), $MachinePrecision] * 0.5), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.9998589899959065

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 93.4

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

      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 98.5%

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

   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 43.8

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

   7. Applied rewrites 43.8%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 98.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 98.0

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

      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 98.6%

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

   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 97.5

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

   7. Applied rewrites 97.5%

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

   if 0.9998589899959065 < (/.f64 (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 76.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{kx}{ky}, kx, \sin ky\right)} \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 88.6%

       \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.05:\\ \;\;\;\;\left(\left(-ky\right) \cdot \sin th\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9998589899959065:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{elif}\;t\_2 \leq 0.9998589899959065:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (* (- th) (sin ky)) (/ -1.0 (hypot (sin ky) (sin kx)))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_2 -0.1)
     t_1
     (if (<= t_2 2e-54)
       (* (/ (sin th) (sin kx)) (sin ky))
       (if (<= t_2 0.9998589899959065)
         t_1
         (* (/ (sin ky) (fma (* (/ kx ky) 0.5) kx (sin ky))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (-th * sin(ky)) * (-1.0 / hypot(sin(ky), sin(kx)));
	double t_2 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_2 <= -0.1) {
		tmp = t_1;
	} else if (t_2 <= 2e-54) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} else if (t_2 <= 0.9998589899959065) {
		tmp = t_1;
	} else {
		tmp = (sin(ky) / fma(((kx / ky) * 0.5), kx, sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(Float64(-th) * sin(ky)) * Float64(-1.0 / hypot(sin(ky), sin(kx))))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = t_1;
	elseif (t_2 <= 2e-54)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	elseif (t_2 <= 0.9998589899959065)
		tmp = t_1;
	else
		tmp = Float64(Float64(sin(ky) / fma(Float64(Float64(kx / ky) * 0.5), kx, sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[((-th) * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $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]}, If[LessEqual[t$95$2, -0.1], t$95$1, If[LessEqual[t$95$2, 2e-54], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9998589899959065], t$95$1, N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(kx / ky), $MachinePrecision] * 0.5), $MachinePrecision] * kx + N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001 or 2.0000000000000001e-54 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.9998589899959065

   1. Initial program 93.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 93.6

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

      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 98.5%

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

   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 44.1

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

   7. Applied rewrites 44.1%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.0000000000000001e-54

   1. Initial program 98.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 98.6

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

      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

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

      1. lower-sin.f64 68.6

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

   7. Applied rewrites 68.6%

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

   if 0.9998589899959065 < (/.f64 (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 76.7%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 88.6

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{1}{2} \cdot \frac{kx}{ky}, kx, \sin ky\right)} \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 88.6%

       \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.9998589899959065:\\ \;\;\;\;\left(\left(-th\right) \cdot \sin ky\right) \cdot \frac{-1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{fma}\left(\frac{kx}{ky} \cdot 0.5, kx, \sin ky\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 5e-6)
   (* (/ (sin th) (sin kx)) (sin ky))
   (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)))) <= 5e-6) {
		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) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 5d-6) 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 tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 5e-6) {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 5e-6:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	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)))) <= 5e-6)
		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)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 5e-6)
		tmp = (sin(th) / sin(kx)) * sin(ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-6], 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 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.00000000000000041e-6

   1. Initial program 95.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 95.4

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

      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. Taylor expanded in ky around 0

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

      1. lower-sin.f64 42.3

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

   7. Applied rewrites 42.3%

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

   if 5.00000000000000041e-6 < (/.f64 (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 85.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 61.2

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

   5. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 5e-6)
   (* (/ (sin 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)))) <= 5e-6) {
		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) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 5d-6) 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 tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 5e-6) {
		tmp = (Math.sin(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)))) <= 5e-6:
		tmp = (math.sin(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)))) <= 5e-6)
		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)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 5e-6)
		tmp = (sin(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], 5e-6], 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 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.00000000000000041e-6

   1. Initial program 95.4%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 42.3

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

   5. Applied rewrites 42.3%

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

   if 5.00000000000000041e-6 < (/.f64 (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 85.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 61.2

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

   5. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 5e-6)
   (* (/ 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)))) <= 5e-6) {
		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)))) <= 5d-6) 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)))) <= 5e-6) {
		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)))) <= 5e-6:
		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)))) <= 5e-6)
		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)))) <= 5e-6)
		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], 5e-6], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.00000000000000041e-6

   1. Initial program 95.4%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 40.0

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

   5. Applied rewrites 40.0%

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

   if 5.00000000000000041e-6 < (/.f64 (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 85.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 61.2

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

   5. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{1}{kx}}\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 2e-18)
   (/ (sin th) (/ (hypot (sin ky) (/ 1.0 (/ 1.0 kx))) (sin ky)))
   (*
    (/
     (sin ky)
     (/
      (sqrt
       (fma (- 1.0 (cos (* 2.0 ky))) 2.0 (* (- 1.0 (cos (* 2.0 kx))) 2.0)))
      2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 2e-18) {
		tmp = sin(th) / (hypot(sin(ky), (1.0 / (1.0 / kx))) / sin(ky));
	} else {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((2.0 * ky))), 2.0, ((1.0 - cos((2.0 * kx))) * 2.0))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 2e-18)
		tmp = Float64(sin(th) / Float64(hypot(sin(ky), Float64(1.0 / Float64(1.0 / kx))) / sin(ky)));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 2.0, Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * 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-18], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(1.0 / N[(1.0 / kx), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $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)) < 2.0000000000000001e-18

   1. Initial program 84.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 84.7

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in kx around 0

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

      1. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if 2.0000000000000001e-18 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   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 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \frac{1}{\frac{1}{kx}}\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, \left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2\right)}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 6.8 \cdot 10^{-51}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 6.8e-51)
   (* (pow th 3.0) -0.16666666666666666)
   (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)))) <= 6.8e-51) {
		tmp = pow(th, 3.0) * -0.16666666666666666;
	} 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)))) <= 6.8d-51) then
        tmp = (th ** 3.0d0) * (-0.16666666666666666d0)
    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)))) <= 6.8e-51) {
		tmp = Math.pow(th, 3.0) * -0.16666666666666666;
	} 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)))) <= 6.8e-51:
		tmp = math.pow(th, 3.0) * -0.16666666666666666
	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)))) <= 6.8e-51)
		tmp = Float64((th ^ 3.0) * -0.16666666666666666);
	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)))) <= 6.8e-51)
		tmp = (th ^ 3.0) * -0.16666666666666666;
	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], 6.8e-51], N[(N[Power[th, 3.0], $MachinePrecision] * -0.16666666666666666), $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))))) < 6.80000000000000005e-51

   1. Initial program 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 3.8

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

   5. Applied rewrites 3.8%

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

   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 3.6%

      \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 18.0%

       \[\leadsto {th}^{3} \cdot -0.16666666666666666 \]

   if 6.80000000000000005e-51 < (/.f64 (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 86.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.2

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

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 6.8 \cdot 10^{-51}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 23.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (sin th))

C

double code(double kx, double ky, double th) {
	return 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(th)
end function

Java

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

Python

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

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in kx around 0

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

   1. lower-sin.f64 21.1

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

5. Applied rewrites 21.1%

   \[\leadsto \color{blue}{\sin th} \]
6. Add Preprocessing

Alternative 16: 12.9% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (fma (* th th) -0.16666666666666666 1.0) th))

C

double code(double kx, double ky, double th) {
	return fma((th * th), -0.16666666666666666, 1.0) * th;
}

Julia

function code(kx, ky, th)
	return Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th)
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

3. Taylor expanded in kx around 0

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

   1. lower-sin.f64 21.1

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

5. Applied rewrites 21.1%

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

6. Taylor expanded in th around 0

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

8. Applied rewrites 9.9%

   \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]
9. Step-by-step derivation

10. Applied rewrites 9.9%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \]
11. Step-by-step derivation

12. Applied rewrites 9.9%

    \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(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)))