Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 10.3s

Alternatives: 20

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

   1. lift-*.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 95.5

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

   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 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 83.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right) \cdot \frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}{th}\right)}^{-1}\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.999:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(ky, ky, t\_2\right)}} \cdot \sin ky\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (pow
          (/
           (*
            (fma (* th th) 0.16666666666666666 1.0)
            (/ (hypot (sin kx) (sin ky)) (sin ky)))
           th)
          -1.0))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_2 t_3)))))
   (if (<= t_4 -0.999)
     (/ (* (sin ky) (sin th)) (sqrt t_3))
     (if (<= t_4 -0.2)
       t_1
       (if (<= t_4 2e-10)
         (* (/ (sin th) (sqrt (fma ky ky t_2))) (sin ky))
         (if (<= t_4 0.995)
           t_1
           (*
            (/
             (sin th)
             (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
            (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(((fma((th * th), 0.16666666666666666, 1.0) * (hypot(sin(kx), sin(ky)) / sin(ky))) / th), -1.0);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.999) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_3);
	} else if (t_4 <= -0.2) {
		tmp = t_1;
	} else if (t_4 <= 2e-10) {
		tmp = (sin(th) / sqrt(fma(ky, ky, t_2))) * sin(ky);
	} else if (t_4 <= 0.995) {
		tmp = t_1;
	} else {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(fma(Float64(th * th), 0.16666666666666666, 1.0) * Float64(hypot(sin(kx), sin(ky)) / sin(ky))) / th) ^ -1.0
	t_2 = sin(kx) ^ 2.0
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.999)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_3));
	elseif (t_4 <= -0.2)
		tmp = t_1;
	elseif (t_4 <= 2e-10)
		tmp = Float64(Float64(sin(th) / sqrt(fma(ky, ky, t_2))) * sin(ky));
	elseif (t_4 <= 0.995)
		tmp = t_1;
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), 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[(N[(N[(N[(th * th), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / th), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.2], t$95$1, If[LessEqual[t$95$4, 2e-10], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(ky * ky + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], t$95$1, N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 85.9

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

   10. Applied rewrites 85.9%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   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. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 56.2%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.0%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.3

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 95.9

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

   10. Applied rewrites 95.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 98.1

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

   12. Applied rewrites 98.1%

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

   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 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 88.8

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

      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 kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 92.7

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

   7. Applied rewrites 92.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.999:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{{\sin ky}^{2}}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right) \cdot \frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}{th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(ky, ky, {\sin kx}^{2}\right)}} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.995:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right) \cdot \frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}{th}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\frac{\mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.999:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(ky, ky, t\_2\right)}} \cdot \sin ky\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (/
          (sin ky)
          (/
           (*
            (fma (* th th) 0.16666666666666666 1.0)
            (hypot (sin kx) (sin ky)))
           th)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_2 t_3)))))
   (if (<= t_4 -0.999)
     (/ (* (sin ky) (sin th)) (sqrt t_3))
     (if (<= t_4 -0.2)
       t_1
       (if (<= t_4 2e-10)
         (* (/ (sin th) (sqrt (fma ky ky t_2))) (sin ky))
         (if (<= t_4 0.995)
           t_1
           (*
            (/
             (sin th)
             (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
            (sin ky))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / ((fma((th * th), 0.16666666666666666, 1.0) * hypot(sin(kx), sin(ky))) / th);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.999) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_3);
	} else if (t_4 <= -0.2) {
		tmp = t_1;
	} else if (t_4 <= 2e-10) {
		tmp = (sin(th) / sqrt(fma(ky, ky, t_2))) * sin(ky);
	} else if (t_4 <= 0.995) {
		tmp = t_1;
	} else {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / Float64(Float64(fma(Float64(th * th), 0.16666666666666666, 1.0) * hypot(sin(kx), sin(ky))) / th))
	t_2 = sin(kx) ^ 2.0
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.999)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_3));
	elseif (t_4 <= -0.2)
		tmp = t_1;
	elseif (t_4 <= 2e-10)
		tmp = Float64(Float64(sin(th) / sqrt(fma(ky, ky, t_2))) * sin(ky));
	elseif (t_4 <= 0.995)
		tmp = t_1;
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), 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[(N[Sin[ky], $MachinePrecision] / N[(N[(N[(N[(th * th), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.2], t$95$1, If[LessEqual[t$95$4, 2e-10], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(ky * ky + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], t$95$1, N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 85.9

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

   10. Applied rewrites 85.9%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   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. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 56.1%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.0%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.3

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 95.9

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

   10. Applied rewrites 95.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 98.1

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

   12. Applied rewrites 98.1%

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

   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 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 88.8

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

      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 kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 92.7

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

   7. Applied rewrites 92.7%

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

Alternative 4: 82.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.999:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(ky, ky, t\_1\right)}} \cdot \sin ky\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \end{array} \end{array} \]

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_1 t_2)))))
   (if (<= t_3 -0.999)
     (/ (* (sin ky) (sin th)) (sqrt t_2))
     (if (<= t_3 -0.2)
       (*
        (* th (* 2.0 (sin ky)))
        (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) (- 1.0 (cos (* 2.0 ky))))))))
       (if (<= t_3 2e-10)
         (* (/ (sin th) (sqrt (fma ky ky t_1))) (sin ky))
         (if (<= t_3 0.995)
           (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
           (*
            (/
             (sin th)
             (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
            (sin ky))))))))

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_1 + t_2));
	double tmp;
	if (t_3 <= -0.999) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_2);
	} else if (t_3 <= -0.2) {
		tmp = (th * (2.0 * sin(ky))) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	} else if (t_3 <= 2e-10) {
		tmp = (sin(th) / sqrt(fma(ky, ky, t_1))) * sin(ky);
	} else if (t_3 <= 0.995) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	}
	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_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.999)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_2));
	elseif (t_3 <= -0.2)
		tmp = Float64(Float64(th * Float64(2.0 * sin(ky))) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - Float64(1.0 - cos(Float64(2.0 * ky))))))));
	elseif (t_3 <= 2e-10)
		tmp = Float64(Float64(sin(th) / sqrt(fma(ky, ky, t_1))) * sin(ky));
	elseif (t_3 <= 0.995)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), 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[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$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[(th * N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-10], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(ky * ky + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 85.9

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

   10. Applied rewrites 85.9%

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

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

   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. *-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 99.3

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. frac-add N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 55.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.0%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.3

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 95.9

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

   10. Applied rewrites 95.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 98.1

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

   12. Applied rewrites 98.1%

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

   if 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.0

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 56.2

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

   9. Applied rewrites 56.2%

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

   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 89.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 88.8

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

      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 kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 92.7

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

   7. Applied rewrites 92.7%

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

Alternative 5: 81.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.999:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin th}{\sqrt{\mathsf{fma}\left(ky, ky, t\_1\right)}} \cdot \sin ky\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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_1 t_2)))))
   (if (<= t_3 -0.999)
     (/ (* (sin ky) (sin th)) (sqrt t_2))
     (if (<= t_3 -0.2)
       (*
        (* th (* 2.0 (sin ky)))
        (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) (- 1.0 (cos (* 2.0 ky))))))))
       (if (<= t_3 2e-10)
         (* (/ (sin th) (sqrt (fma ky ky t_1))) (sin ky))
         (if (<= t_3 0.995)
           (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
           (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_1 + t_2));
	double tmp;
	if (t_3 <= -0.999) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_2);
	} else if (t_3 <= -0.2) {
		tmp = (th * (2.0 * sin(ky))) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	} else if (t_3 <= 2e-10) {
		tmp = (sin(th) / sqrt(fma(ky, ky, t_1))) * sin(ky);
	} else if (t_3 <= 0.995) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = 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_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.999)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_2));
	elseif (t_3 <= -0.2)
		tmp = Float64(Float64(th * Float64(2.0 * sin(ky))) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - Float64(1.0 - cos(Float64(2.0 * ky))))))));
	elseif (t_3 <= 2e-10)
		tmp = Float64(Float64(sin(th) / sqrt(fma(ky, ky, t_1))) * sin(ky));
	elseif (t_3 <= 0.995)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = 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$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[(th * N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-10], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(ky * ky + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 85.9

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

   10. Applied rewrites 85.9%

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

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

   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. *-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 99.3

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. frac-add N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 55.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.0%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.3

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 95.9

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

   10. Applied rewrites 95.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 98.1

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

   12. Applied rewrites 98.1%

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

   if 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.0

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 56.2

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

   9. Applied rewrites 56.2%

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

   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 89.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

Alternative 6: 81.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_1 + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.999:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, t\_1\right)}}\\ \mathbf{elif}\;t\_3 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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_1 t_2)))))
   (if (<= t_3 -0.999)
     (/ (* (sin ky) (sin th)) (sqrt t_2))
     (if (<= t_3 -0.2)
       (*
        (* th (* 2.0 (sin ky)))
        (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) (- 1.0 (cos (* 2.0 ky))))))))
       (if (<= t_3 2e-10)
         (* (sin th) (/ (sin ky) (sqrt (fma ky ky t_1))))
         (if (<= t_3 0.995)
           (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
           (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_1 + t_2));
	double tmp;
	if (t_3 <= -0.999) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_2);
	} else if (t_3 <= -0.2) {
		tmp = (th * (2.0 * sin(ky))) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	} else if (t_3 <= 2e-10) {
		tmp = sin(th) * (sin(ky) / sqrt(fma(ky, ky, t_1)));
	} else if (t_3 <= 0.995) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = 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_1 + t_2)))
	tmp = 0.0
	if (t_3 <= -0.999)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_2));
	elseif (t_3 <= -0.2)
		tmp = Float64(Float64(th * Float64(2.0 * sin(ky))) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - Float64(1.0 - cos(Float64(2.0 * ky))))))));
	elseif (t_3 <= 2e-10)
		tmp = Float64(sin(th) * Float64(sin(ky) / sqrt(fma(ky, ky, t_1))));
	elseif (t_3 <= 0.995)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = 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$1 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[(th * N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e-10], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * ky + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.995], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 85.9

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

   10. Applied rewrites 85.9%

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

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

   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. *-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 99.3

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. frac-add N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 55.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.0%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.3

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 95.9

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

   10. Applied rewrites 95.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 97.8

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

   12. Applied rewrites 97.8%

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

   if 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.0

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 56.2

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

   9. Applied rewrites 56.2%

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

   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 89.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

Alternative 7: 69.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.999:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_1}}\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.999)
     (/ (* (sin ky) (sin th)) (sqrt t_1))
     (if (<= t_2 -0.2)
       (*
        (* th (* 2.0 (sin ky)))
        (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) (- 1.0 (cos (* 2.0 ky))))))))
       (if (<= t_2 2e-10)
         (* (sin ky) (/ (sin th) (sin kx)))
         (if (<= t_2 0.995)
           (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_1);
	} else if (t_2 <= -0.2) {
		tmp = (th * (2.0 * sin(ky))) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	} else if (t_2 <= 2e-10) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (t_2 <= 0.995) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.999) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sqrt(t_1);
	} else if (t_2 <= -0.2) {
		tmp = (th * (2.0 * Math.sin(ky))) * Math.sqrt((0.5 / (1.0 - (Math.cos((2.0 * kx)) - (1.0 - Math.cos((2.0 * ky)))))));
	} else if (t_2 <= 2e-10) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (t_2 <= 0.995) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	tmp = 0
	if t_2 <= -0.999:
		tmp = (math.sin(ky) * math.sin(th)) / math.sqrt(t_1)
	elif t_2 <= -0.2:
		tmp = (th * (2.0 * math.sin(ky))) * math.sqrt((0.5 / (1.0 - (math.cos((2.0 * kx)) - (1.0 - math.cos((2.0 * ky)))))))
	elif t_2 <= 2e-10:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif t_2 <= 0.995:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.999)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_1));
	elseif (t_2 <= -0.2)
		tmp = Float64(Float64(th * Float64(2.0 * sin(ky))) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - Float64(1.0 - cos(Float64(2.0 * ky))))))));
	elseif (t_2 <= 2e-10)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (t_2 <= 0.995)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	tmp = 0.0;
	if (t_2 <= -0.999)
		tmp = (sin(ky) * sin(th)) / sqrt(t_1);
	elseif (t_2 <= -0.2)
		tmp = (th * (2.0 * sin(ky))) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	elseif (t_2 <= 2e-10)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (t_2 <= 0.995)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], N[(N[(th * N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-10], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 85.9

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

   10. Applied rewrites 85.9%

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

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

   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. *-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 99.3

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. frac-add N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 55.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.0%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.0

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 56.2

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

   9. Applied rewrites 56.2%

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

   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 89.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

Alternative 8: 64.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := 1 - \cos \left(2 \cdot ky\right)\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_2} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - t\_2\right)}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (- 1.0 (cos (* 2.0 ky)))))
   (if (<= t_1 -0.999)
     (/ (* (sin ky) (sin th)) (* (sqrt t_2) (sqrt 0.5)))
     (if (<= t_1 -0.2)
       (*
        (* th (* 2.0 (sin ky)))
        (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) t_2)))))
       (if (<= t_1 2e-10)
         (* (sin ky) (/ (sin th) (sin kx)))
         (if (<= t_1 0.995)
           (/ (sin ky) (/ (hypot (sin kx) (sin ky)) th))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = 1.0 - cos((2.0 * ky));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(ky) * sin(th)) / (sqrt(t_2) * sqrt(0.5));
	} else if (t_1 <= -0.2) {
		tmp = (th * (2.0 * sin(ky))) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - t_2))));
	} else if (t_1 <= 2e-10) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (t_1 <= 0.995) {
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_2 = 1.0 - Math.cos((2.0 * ky));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (Math.sin(ky) * Math.sin(th)) / (Math.sqrt(t_2) * Math.sqrt(0.5));
	} else if (t_1 <= -0.2) {
		tmp = (th * (2.0 * Math.sin(ky))) * Math.sqrt((0.5 / (1.0 - (Math.cos((2.0 * kx)) - t_2))));
	} else if (t_1 <= 2e-10) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (t_1 <= 0.995) {
		tmp = Math.sin(ky) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_2 = 1.0 - math.cos((2.0 * ky))
	tmp = 0
	if t_1 <= -0.999:
		tmp = (math.sin(ky) * math.sin(th)) / (math.sqrt(t_2) * math.sqrt(0.5))
	elif t_1 <= -0.2:
		tmp = (th * (2.0 * math.sin(ky))) * math.sqrt((0.5 / (1.0 - (math.cos((2.0 * kx)) - t_2))))
	elif t_1 <= 2e-10:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif t_1 <= 0.995:
		tmp = math.sin(ky) / (math.hypot(math.sin(kx), math.sin(ky)) / th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(1.0 - cos(Float64(2.0 * ky)))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(ky) * sin(th)) / Float64(sqrt(t_2) * sqrt(0.5)));
	elseif (t_1 <= -0.2)
		tmp = Float64(Float64(th * Float64(2.0 * sin(ky))) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - t_2)))));
	elseif (t_1 <= 2e-10)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (t_1 <= 0.995)
		tmp = Float64(sin(ky) / Float64(hypot(sin(kx), sin(ky)) / th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_2 = 1.0 - cos((2.0 * ky));
	tmp = 0.0;
	if (t_1 <= -0.999)
		tmp = (sin(ky) * sin(th)) / (sqrt(t_2) * sqrt(0.5));
	elseif (t_1 <= -0.2)
		tmp = (th * (2.0 * sin(ky))) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - t_2))));
	elseif (t_1 <= 2e-10)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (t_1 <= 0.995)
		tmp = sin(ky) / (hypot(sin(kx), sin(ky)) / th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$2], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], N[(N[(th * N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-10], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / th), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 84.6

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

   7. Applied rewrites 84.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. cos-diff N/A

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

      9. cos-sin-sum N/A

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

      10. count-2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift--.f64 70.3

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

   9. Applied rewrites 70.3%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 71.1

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

   12. Applied rewrites 71.1%

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

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

   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. *-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 99.3

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

      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 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. frac-add N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 55.0%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.0%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.0

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in th around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 56.2

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

   9. Applied rewrites 56.2%

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

   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 89.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

Alternative 9: 64.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := 1 - \cos \left(2 \cdot ky\right)\\ t_3 := \left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - t\_2\right)}}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_2} \cdot \sqrt{0.5}}\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (- 1.0 (cos (* 2.0 ky))))
        (t_3
         (*
          (* th (* 2.0 (sin ky)))
          (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) t_2)))))))
   (if (<= t_1 -0.999)
     (/ (* (sin ky) (sin th)) (* (sqrt t_2) (sqrt 0.5)))
     (if (<= t_1 -0.2)
       t_3
       (if (<= t_1 2e-10)
         (* (sin ky) (/ (sin th) (sin kx)))
         (if (<= t_1 0.995) t_3 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = 1.0 - cos((2.0 * ky));
	double t_3 = (th * (2.0 * sin(ky))) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - t_2))));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(ky) * sin(th)) / (sqrt(t_2) * sqrt(0.5));
	} else if (t_1 <= -0.2) {
		tmp = t_3;
	} else if (t_1 <= 2e-10) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else if (t_1 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    t_2 = 1.0d0 - cos((2.0d0 * ky))
    t_3 = (th * (2.0d0 * sin(ky))) * sqrt((0.5d0 / (1.0d0 - (cos((2.0d0 * kx)) - t_2))))
    if (t_1 <= (-0.999d0)) then
        tmp = (sin(ky) * sin(th)) / (sqrt(t_2) * sqrt(0.5d0))
    else if (t_1 <= (-0.2d0)) then
        tmp = t_3
    else if (t_1 <= 2d-10) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else if (t_1 <= 0.995d0) then
        tmp = t_3
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_2 = 1.0 - Math.cos((2.0 * ky));
	double t_3 = (th * (2.0 * Math.sin(ky))) * Math.sqrt((0.5 / (1.0 - (Math.cos((2.0 * kx)) - t_2))));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (Math.sin(ky) * Math.sin(th)) / (Math.sqrt(t_2) * Math.sqrt(0.5));
	} else if (t_1 <= -0.2) {
		tmp = t_3;
	} else if (t_1 <= 2e-10) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else if (t_1 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_2 = 1.0 - math.cos((2.0 * ky))
	t_3 = (th * (2.0 * math.sin(ky))) * math.sqrt((0.5 / (1.0 - (math.cos((2.0 * kx)) - t_2))))
	tmp = 0
	if t_1 <= -0.999:
		tmp = (math.sin(ky) * math.sin(th)) / (math.sqrt(t_2) * math.sqrt(0.5))
	elif t_1 <= -0.2:
		tmp = t_3
	elif t_1 <= 2e-10:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	elif t_1 <= 0.995:
		tmp = t_3
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(1.0 - cos(Float64(2.0 * ky)))
	t_3 = Float64(Float64(th * Float64(2.0 * sin(ky))) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - t_2)))))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(ky) * sin(th)) / Float64(sqrt(t_2) * sqrt(0.5)));
	elseif (t_1 <= -0.2)
		tmp = t_3;
	elseif (t_1 <= 2e-10)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	elseif (t_1 <= 0.995)
		tmp = t_3;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_2 = 1.0 - cos((2.0 * ky));
	t_3 = (th * (2.0 * sin(ky))) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - t_2))));
	tmp = 0.0;
	if (t_1 <= -0.999)
		tmp = (sin(ky) * sin(th)) / (sqrt(t_2) * sqrt(0.5));
	elseif (t_1 <= -0.2)
		tmp = t_3;
	elseif (t_1 <= 2e-10)
		tmp = sin(ky) * (sin(th) / sin(kx));
	elseif (t_1 <= 0.995)
		tmp = t_3;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(th * N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$2], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$3, If[LessEqual[t$95$1, 2e-10], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.995], t$95$3, N[Sin[th], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 90.8%

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

   3. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.6

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

   5. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 84.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 84.6

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

   7. Applied rewrites 84.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. cos-diff N/A

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

      9. cos-sin-sum N/A

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

      10. count-2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift--.f64 70.3

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

   9. Applied rewrites 70.3%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 71.1

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

   12. Applied rewrites 71.1%

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

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   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. *-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 99.2

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

      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 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-hypot.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. frac-add N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sqrt-div N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 55.3%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 99.0%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 62.2

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

   5. Applied rewrites 62.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 62.2

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

   7. Applied rewrites 62.2%

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

   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 89.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

Alternative 10: 58.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.2d0)) then
        tmp = (sin(ky) * sin(th)) / (sqrt((1.0d0 - cos((2.0d0 * ky)))) * sqrt(0.5d0))
    else if (t_1 <= 0.5d0) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = (Math.sin(ky) * Math.sin(th)) / (Math.sqrt((1.0 - Math.cos((2.0 * ky)))) * Math.sqrt(0.5));
	} else if (t_1 <= 0.5) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.2:
		tmp = (math.sin(ky) * math.sin(th)) / (math.sqrt((1.0 - math.cos((2.0 * ky)))) * math.sqrt(0.5))
	elif t_1 <= 0.5:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = (sin(ky) * sin(th)) / (sqrt((1.0 - cos((2.0 * ky)))) * sqrt(0.5));
	elseif (t_1 <= 0.5)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.6%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.6

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

   5. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 49.5

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 49.5

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

   7. Applied rewrites 49.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. sin-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. cos-diff N/A

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

      9. cos-sin-sum N/A

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

      10. count-2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. lift--.f64 41.6

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

   9. Applied rewrites 41.6%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-sqrt.f64 49.3

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

   12. Applied rewrites 49.3%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.5

   1. Initial program 99.0%

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

   3. Taylor expanded in ky around 0

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

      1. lower-sin.f64 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin kx}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      6. lower-/.f64 55.9

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   7. Applied rewrites 55.9%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

   if 0.5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 68.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.999:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 0.5, kx \cdot kx\right)}}\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.999)
     (/
      (* (sin ky) (sin th))
      (sqrt (fma (- 1.0 (cos (* 2.0 ky))) 0.5 (* kx kx))))
     (if (<= t_1 0.5) (* (sin ky) (/ (sin th) (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.999) {
		tmp = (sin(ky) * sin(th)) / sqrt(fma((1.0 - cos((2.0 * ky))), 0.5, (kx * kx)));
	} else if (t_1 <= 0.5) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.999)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(fma(Float64(1.0 - cos(Float64(2.0 * ky))), 0.5, Float64(kx * kx))));
	elseif (t_1 <= 0.5)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.999], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.998999999999999999

   1. Initial program 90.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 86.6

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 86.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{kx \cdot kx + {\sin ky}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{kx \cdot kx + {\sin ky}^{2}}}} \]

      5. lower-*.f64 84.6

         \[\leadsto \frac{\color{blue}{\sin ky \cdot \sin th}}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{kx \cdot kx + {\sin ky}^{2}}}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + kx \cdot kx}}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + kx \cdot kx}} \]

      9. pow2 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + kx \cdot kx}} \]

      10. lower-fma.f64 84.6

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\sin ky, \sin ky, kx \cdot kx\right)}}} \]

   7. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\sin ky, \sin ky, kx \cdot kx\right)}}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky + kx \cdot kx}}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + kx \cdot kx}} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + kx \cdot kx}} \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + kx \cdot kx}} \]

      5. div-inv N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \frac{1}{2}} + kx \cdot kx}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot \color{blue}{\frac{1}{2}} + kx \cdot kx}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right), \frac{1}{2}, kx \cdot kx\right)}}} \]

      8. cos-diff N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(ky + ky\right), \frac{1}{2}, kx \cdot kx\right)}} \]

      9. cos-sin-sum N/A

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\color{blue}{1} - \cos \left(ky + ky\right), \frac{1}{2}, kx \cdot kx\right)}} \]

      10. count-2 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \]

      12. lift-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot ky\right)}, \frac{1}{2}, kx \cdot kx\right)}} \]

      13. lift--.f64 70.3

          \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot ky\right)}, 0.5, kx \cdot kx\right)}} \]

   9. Applied rewrites 70.3%

      \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 0.5, kx \cdot kx\right)}}} \]

   if -0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.5

   1. Initial program 99.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 43.1

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 43.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin kx} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin kx}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      6. lower-/.f64 43.1

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

   if 0.5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 68.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 45.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.5:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.5)
   (* (sin ky) (/ (sin th) (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.5) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.5d0) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.5) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.5:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.5)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.5)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.5], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.5

   1. Initial program 96.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 33.5

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 33.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin kx} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sin kx}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

      6. lower-/.f64 33.5

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sin kx}} \]

   7. Applied rewrites 33.5%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sin kx}} \]

   if 0.5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 68.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 44.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-10)
   (/ (sin th) (/ (sin kx) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-10) {
		tmp = sin(th) / (sin(kx) / 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(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 2d-10) then
        tmp = sin(th) / (sin(kx) / 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(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 2e-10) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 2e-10:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-10)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-10)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-10], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / 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))))) < 2.00000000000000007e-10

   1. Initial program 96.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. *-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 96.8

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      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 99.6

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

      2. lower-sin.f64 33.0

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sin kx}}{ky}} \]

   7. Applied rewrites 33.0%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

   if 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 44.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-10)
   (* (sin th) (/ ky (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 2e-10) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 2d-10) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 2e-10) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 2e-10:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-10)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-10)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-10], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 96.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sin kx} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{ky}{\sin kx}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin th \cdot \frac{ky}{\sin kx}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin th} \cdot \frac{ky}{\sin kx} \]

      5. lower-/.f64 N/A

         \[\leadsto \sin th \cdot \color{blue}{\frac{ky}{\sin kx}} \]

      6. lower-sin.f64 33.0

         \[\leadsto \sin th \cdot \frac{ky}{\color{blue}{\sin kx}} \]

   5. Applied rewrites 33.0%

      \[\leadsto \color{blue}{\sin th \cdot \frac{ky}{\sin kx}} \]

   if 2.00000000000000007e-10 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 5e-6)
   (*
    (/
     (sin th)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin ky))
   (/
    (sin th)
    (/
     (/
      (sqrt (* 2.0 (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx))))))
      2.0)
     (sin ky)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 5e-6) {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	} else {
		tmp = sin(th) / ((sqrt((2.0 * ((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx)))))) / 2.0) / sin(ky));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 5e-6)
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky));
	else
		tmp = Float64(sin(th) / Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx)))))) / 2.0) / sin(ky)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 5e-6], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[(N[Sqrt[N[(2.0 * N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 91.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 91.2

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      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 99.7

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin ky \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin ky \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin ky \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin ky \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin ky \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin ky \]

      6. lower-*.f64 99.6

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin ky \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin ky \]

   if 5.00000000000000041e-6 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   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. *-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 99.4

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      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 99.4

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{\sin ky}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky}} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}}{\sin ky}} \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}}{\sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}}{\sin ky}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}}{\sin ky}} \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}}{\sin ky}} \]

      8. frac-add N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}}{\sin ky}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}}{\sin ky}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}}{\sin ky}} \]

      11. sqrt-div N/A

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}}{\sin ky}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{\color{blue}{4}}}}{\sin ky}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\color{blue}{2}}}{\sin ky}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{2}}}{\sin ky}} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}}{\sin ky}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2 + 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)}}}{2}}{\sin ky}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)} + 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)}}{2}}{\sin ky}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right) + \color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)}}}{2}}{\sin ky}} \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}}{2}}{\sin ky}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}}{2}}{\sin ky}} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right)} + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}{\sin ky}} \]

      7. cos-sin-sum N/A

         \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\color{blue}{\left(\cos ky \cdot \cos ky + \sin ky \cdot \sin ky\right)} - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}{\sin ky}} \]

      8. cos-diff N/A

         \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\color{blue}{\cos \left(ky - ky\right)} - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}{\sin ky}} \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\cos \left(ky - ky\right) - \color{blue}{\cos \left(2 \cdot ky\right)}\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}{\sin ky}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(2 \cdot ky\right)}\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}{\sin ky}} \]

      11. count-2 N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\cos \left(ky - ky\right) - \cos \color{blue}{\left(ky + ky\right)}\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}{\sin ky}} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right)}\right)}}{2}}{\sin ky}} \]

      13. cos-sin-sum N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\color{blue}{\left(\cos kx \cdot \cos kx + \sin kx \cdot \sin kx\right)} - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}{\sin ky}} \]

      14. cos-diff N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\color{blue}{\cos \left(kx - kx\right)} - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}{\sin ky}} \]

      15. lift-cos.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \color{blue}{\cos \left(2 \cdot kx\right)}\right)\right)}}{2}}{\sin ky}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \color{blue}{\left(2 \cdot kx\right)}\right)\right)}}{2}}{\sin ky}} \]

      17. count-2 N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{2 \cdot \left(\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) + \left(\cos \left(kx - kx\right) - \cos \color{blue}{\left(kx + kx\right)}\right)\right)}}{2}}{\sin ky}} \]

   8. Applied rewrites 99.1%

      \[\leadsto \frac{\sin th}{\frac{\frac{\color{blue}{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}}{2}}{\sin ky}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sin th\right) \cdot \sin ky\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 5e-6)
   (*
    (/
     (sin th)
     (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
    (sin ky))
   (*
    (* (* 2.0 (sin th)) (sin ky))
    (sqrt (/ 0.5 (- 1.0 (- (cos (* 2.0 kx)) (- 1.0 (cos (* 2.0 ky))))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 5e-6) {
		tmp = (sin(th) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(ky);
	} else {
		tmp = ((2.0 * sin(th)) * sin(ky)) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 5e-6)
		tmp = Float64(Float64(sin(th) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(ky));
	else
		tmp = Float64(Float64(Float64(2.0 * sin(th)) * sin(ky)) * sqrt(Float64(0.5 / Float64(1.0 - Float64(cos(Float64(2.0 * kx)) - Float64(1.0 - cos(Float64(2.0 * ky))))))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 5e-6], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 91.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 91.2

         \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

      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 99.7

          \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin ky \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin ky \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot kx}\right)} \cdot \sin ky \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\left(\frac{-1}{6} \cdot {kx}^{2} + 1\right)} \cdot kx\right)} \cdot \sin ky \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right)} \cdot kx\right)} \cdot \sin ky \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin ky \]

      6. lower-*.f64 99.6

         \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right)} \cdot \sin ky \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin ky \]

   if 5.00000000000000041e-6 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   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. *-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 99.4

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      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 99.4

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]
   5. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}}{\sin ky}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky}} \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + \sin kx \cdot \sin kx}}{\sin ky}} \]

      4. sin-mult N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + \sin kx \cdot \sin kx}}{\sin ky}} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}}{\sin ky}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}}{\sin ky}} \]

      7. sin-mult N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}}{\sin ky}} \]

      8. frac-add N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}}{\sin ky}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}}{\sin ky}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\frac{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}}{\sin ky}} \]

      11. sqrt-div N/A

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}}{\sin ky}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{\color{blue}{4}}}}{\sin ky}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\sin th}{\frac{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\color{blue}{2}}}{\sin ky}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\sin th}{\frac{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{2}}}{\sin ky}} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot ky\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}}{\sin ky}} \]

   7. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{2 \cdot \left(\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot \sin th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot \sin th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      3. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin th\right) \cdot \sin ky\right)} \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \sin th\right) \cdot \sin ky\right)} \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \sin th\right)} \cdot \sin ky\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\sin th}\right) \cdot \sin ky\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      8. lower-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \sin th\right) \cdot \color{blue}{\sin ky}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(2 \cdot \sin th\right) \cdot \sin ky\right) \cdot \color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      10. distribute-lft-out N/A

          \[\leadsto \left(\left(2 \cdot \sin th\right) \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      11. associate-/r* N/A

          \[\leadsto \left(\left(2 \cdot \sin th\right) \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(2 \cdot \sin th\right) \cdot \sin ky\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \sin th\right) \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

   9. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin th\right) \cdot \sin ky\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 31.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 3.4 \cdot 10^{-43}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 3.4e-43)
   (* (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(kx), 2.0) + pow(sin(ky), 2.0)))) <= 3.4e-43) {
		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(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 3.4d-43) 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(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 3.4e-43) {
		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(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 3.4e-43:
		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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 3.4e-43)
		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(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 3.4e-43)
		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[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.4e-43], 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))))) < 3.4000000000000001e-43

   1. Initial program 97.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.4%

      \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 14.1%

       \[\leadsto {th}^{3} \cdot -0.16666666666666666 \]

   if 3.4000000000000001e-43 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 92.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 59.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (hypot (sin ky) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-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. Add Preprocessing

Alternative 19: 24.3% 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 95.5%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 23.9

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 23.9%

   \[\leadsto \color{blue}{\sin th} \]
6. Add Preprocessing

Alternative 20: 13.3% 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 95.5%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 23.9

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 23.9%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 11.9%

   \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]
9. Step-by-step derivation

10. Applied rewrites 11.9%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(th \cdot th\right), th, th\right) \]
11. Step-by-step derivation

12. Applied rewrites 11.9%

    \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(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)))