Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 12.5s

Alternatives: 31

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

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 96.6%

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

   1. lift-*.f64 N/A

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

   2. *-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 96.6

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

   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: 81.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + t\_1}}\\ t_4 := {\left(\frac{\mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right) \cdot \frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}{th}\right)}^{-1}\\ \mathbf{if}\;t\_3 \leq -0.998:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.001:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2 + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.96:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 t_1))))
        (t_4
         (pow
          (/
           (*
            (fma (* th th) 0.16666666666666666 1.0)
            (/ (hypot (sin kx) (sin ky)) (sin ky)))
           th)
          -1.0)))
   (if (<= t_3 -0.998)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_3 -0.2)
       t_4
       (if (<= t_3 0.001)
         (* (/ (sin ky) (sqrt (+ t_2 (* ky ky)))) (sin th))
         (if (<= t_3 0.96) t_4 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + t_1));
	double t_4 = pow(((fma((th * th), 0.16666666666666666, 1.0) * (hypot(sin(kx), sin(ky)) / sin(ky))) / th), -1.0);
	double tmp;
	if (t_3 <= -0.998) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 0.001) {
		tmp = (sin(ky) / sqrt((t_2 + (ky * ky)))) * sin(th);
	} else if (t_3 <= 0.96) {
		tmp = t_4;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + t_1)))
	t_4 = Float64(Float64(fma(Float64(th * th), 0.16666666666666666, 1.0) * Float64(hypot(sin(kx), sin(ky)) / sin(ky))) / th) ^ -1.0
	tmp = 0.0
	if (t_3 <= -0.998)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 0.001)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(t_2 + Float64(ky * ky)))) * sin(th));
	elseif (t_3 <= 0.96)
		tmp = t_4;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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]}, If[LessEqual[t$95$3, -0.998], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], t$95$4, If[LessEqual[t$95$3, 0.001], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.96], t$95$4, 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))))) < -0.998

   1. Initial program 88.9%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

   if -0.998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   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 47.0%

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

   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))))) < 1e-3

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

   if 0.95999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.998:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right) \cdot \frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}{th}\right)}^{-1}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.96:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right) \cdot \frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}{th}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + t\_1}}\\ t_4 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;t\_3 \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0.06:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2 + ky \cdot ky}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.96:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 t_1))))
        (t_4 (/ (* (sin ky) th) (hypot (sin ky) (sin kx)))))
   (if (<= t_3 -0.9995)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_1))) (sin th))
     (if (<= t_3 -0.2)
       t_4
       (if (<= t_3 0.06)
         (* (/ (sin ky) (sqrt (+ t_2 (* ky ky)))) (sin th))
         (if (<= t_3 0.96) t_4 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + t_1));
	double t_4 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double tmp;
	if (t_3 <= -0.9995) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 0.06) {
		tmp = (sin(ky) / sqrt((t_2 + (ky * ky)))) * sin(th);
	} else if (t_3 <= 0.96) {
		tmp = t_4;
	} 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.pow(Math.sin(kx), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((t_2 + t_1));
	double t_4 = (Math.sin(ky) * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (t_3 <= -0.9995) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_1))) * Math.sin(th);
	} else if (t_3 <= -0.2) {
		tmp = t_4;
	} else if (t_3 <= 0.06) {
		tmp = (Math.sin(ky) / Math.sqrt((t_2 + (ky * ky)))) * Math.sin(th);
	} else if (t_3 <= 0.96) {
		tmp = t_4;
	} 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.pow(math.sin(kx), 2.0)
	t_3 = math.sin(ky) / math.sqrt((t_2 + t_1))
	t_4 = (math.sin(ky) * th) / math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if t_3 <= -0.9995:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_1))) * math.sin(th)
	elif t_3 <= -0.2:
		tmp = t_4
	elif t_3 <= 0.06:
		tmp = (math.sin(ky) / math.sqrt((t_2 + (ky * ky)))) * math.sin(th)
	elif t_3 <= 0.96:
		tmp = t_4
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + t_1)))
	t_4 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	tmp = 0.0
	if (t_3 <= -0.9995)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_1))) * sin(th));
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 0.06)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(t_2 + Float64(ky * ky)))) * sin(th));
	elseif (t_3 <= 0.96)
		tmp = t_4;
	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(kx) ^ 2.0;
	t_3 = sin(ky) / sqrt((t_2 + t_1));
	t_4 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (t_3 <= -0.9995)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_1))) * sin(th);
	elseif (t_3 <= -0.2)
		tmp = t_4;
	elseif (t_3 <= 0.06)
		tmp = (sin(ky) / sqrt((t_2 + (ky * ky)))) * sin(th);
	elseif (t_3 <= 0.96)
		tmp = t_4;
	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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.9995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], t$95$4, If[LessEqual[t$95$3, 0.06], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.96], t$95$4, 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))))) < -0.99950000000000006

   1. Initial program 88.9%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

   if -0.99950000000000006 < (/.f64 (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 0.059999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   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. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 46.2

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

   7. Applied rewrites 46.2%

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

   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))))) < 0.059999999999999998

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if 0.95999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.4

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

   5. Applied rewrites 92.4%

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

Alternative 4: 82.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.06:\\ \;\;\;\;\frac{-1}{\frac{-1}{ky} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.96:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin ky) (sin kx))))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.9995)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_2))) (sin th))
     (if (<= t_3 -0.05)
       t_1
       (if (<= t_3 0.06)
         (* (/ -1.0 (* (/ -1.0 ky) (hypot (sin kx) (sin ky)))) (sin th))
         (if (<= t_3 0.96) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.9995) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	} else if (t_3 <= -0.05) {
		tmp = t_1;
	} else if (t_3 <= 0.06) {
		tmp = (-1.0 / ((-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th);
	} else if (t_3 <= 0.96) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) * th) / Math.hypot(Math.sin(ky), Math.sin(kx));
	double t_2 = Math.pow(Math.sin(ky), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.9995) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + t_2))) * Math.sin(th);
	} else if (t_3 <= -0.05) {
		tmp = t_1;
	} else if (t_3 <= 0.06) {
		tmp = (-1.0 / ((-1.0 / ky) * Math.hypot(Math.sin(kx), Math.sin(ky)))) * Math.sin(th);
	} else if (t_3 <= 0.96) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) * th) / math.hypot(math.sin(ky), math.sin(kx))
	t_2 = math.pow(math.sin(ky), 2.0)
	t_3 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_2))
	tmp = 0
	if t_3 <= -0.9995:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + t_2))) * math.sin(th)
	elif t_3 <= -0.05:
		tmp = t_1
	elif t_3 <= 0.06:
		tmp = (-1.0 / ((-1.0 / ky) * math.hypot(math.sin(kx), math.sin(ky)))) * math.sin(th)
	elif t_3 <= 0.96:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.9995)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_2))) * sin(th));
	elseif (t_3 <= -0.05)
		tmp = t_1;
	elseif (t_3 <= 0.06)
		tmp = Float64(Float64(-1.0 / Float64(Float64(-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th));
	elseif (t_3 <= 0.96)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	t_2 = sin(ky) ^ 2.0;
	t_3 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_2));
	tmp = 0.0;
	if (t_3 <= -0.9995)
		tmp = (sin(ky) / sqrt(((kx * kx) + t_2))) * sin(th);
	elseif (t_3 <= -0.05)
		tmp = t_1;
	elseif (t_3 <= 0.06)
		tmp = (-1.0 / ((-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th);
	elseif (t_3 <= 0.96)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.9995], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.05], t$95$1, If[LessEqual[t$95$3, 0.06], N[(N[(-1.0 / N[(N[(-1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.96], t$95$1, 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))))) < -0.99950000000000006

   1. Initial program 88.9%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 88.9

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

   5. Applied rewrites 88.9%

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

   if -0.99950000000000006 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 0.059999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   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. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 46.3

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

   7. Applied rewrites 46.3%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.059999999999999998

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.5

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 98.2

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

   9. Applied rewrites 98.2%

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

   if 0.95999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.06:\\ \;\;\;\;\frac{-1}{\frac{-1}{ky} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.96:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;t\_1 \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), kx \cdot kx, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.06:\\ \;\;\;\;\frac{-1}{\frac{-1}{ky} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.96:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (/ (* (sin ky) th) (hypot (sin ky) (sin kx)))))
   (if (<= t_1 -0.9995)
     (*
      (/
       (sin ky)
       (/
        (sqrt
         (fma
          (- 1.0 (cos (* ky 2.0)))
          2.0
          (*
           (*
            (fma
             (fma 0.17777777777777778 (* kx kx) -1.3333333333333333)
             (* kx kx)
             4.0)
            kx)
           kx)))
        2.0))
      (sin th))
     (if (<= t_1 -0.05)
       t_2
       (if (<= t_1 0.06)
         (* (/ -1.0 (* (/ -1.0 ky) (hypot (sin kx) (sin ky)))) (sin th))
         (if (<= t_1 0.96) t_2 (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 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double tmp;
	if (t_1 <= -0.9995) {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((ky * 2.0))), 2.0, ((fma(fma(0.17777777777777778, (kx * kx), -1.3333333333333333), (kx * kx), 4.0) * kx) * kx))) / 2.0)) * sin(th);
	} else if (t_1 <= -0.05) {
		tmp = t_2;
	} else if (t_1 <= 0.06) {
		tmp = (-1.0 / ((-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th);
	} else if (t_1 <= 0.96) {
		tmp = t_2;
	} 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))))
	t_2 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	tmp = 0.0
	if (t_1 <= -0.9995)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(ky * 2.0))), 2.0, Float64(Float64(fma(fma(0.17777777777777778, Float64(kx * kx), -1.3333333333333333), Float64(kx * kx), 4.0) * kx) * kx))) / 2.0)) * sin(th));
	elseif (t_1 <= -0.05)
		tmp = t_2;
	elseif (t_1 <= 0.06)
		tmp = Float64(Float64(-1.0 / Float64(Float64(-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th));
	elseif (t_1 <= 0.96)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9995], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(0.17777777777777778 * N[(kx * kx), $MachinePrecision] + -1.3333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 4.0), $MachinePrecision] * kx), $MachinePrecision] * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$2, If[LessEqual[t$95$1, 0.06], N[(N[(-1.0 / N[(N[(-1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.96], t$95$2, N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99950000000000006

   1. Initial program 88.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{{kx}^{2} \cdot \left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right)}\right)}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot {kx}^{2}}\right)}}{2}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(kx \cdot kx\right)}\right)}}{2}} \cdot \sin th \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right)} \cdot kx\right)}}{2}} \cdot \sin th \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\color{blue}{\left({kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right) + 4\right)} \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\left(\color{blue}{\left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right) \cdot {kx}^{2}} + 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\color{blue}{\mathsf{fma}\left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}, {kx}^{2}, 4\right)} \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      9. sub-neg N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\color{blue}{\frac{8}{45} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\frac{8}{45} \cdot {kx}^{2} + \color{blue}{\frac{-4}{3}}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{8}{45}, {kx}^{2}, \frac{-4}{3}\right)}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, \color{blue}{kx \cdot kx}, \frac{-4}{3}\right), {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, \color{blue}{kx \cdot kx}, \frac{-4}{3}\right), {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, kx \cdot kx, \frac{-4}{3}\right), \color{blue}{kx \cdot kx}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      15. lower-*.f64 60.2

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), \color{blue}{kx \cdot kx}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), kx \cdot kx, 4\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

   if -0.99950000000000006 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 0.059999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   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. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 46.3

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

   7. Applied rewrites 46.3%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.059999999999999998

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.5

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 98.2

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

   9. Applied rewrites 98.2%

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

   if 0.95999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), kx \cdot kx, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.06:\\ \;\;\;\;\frac{-1}{\frac{-1}{ky} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.96:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_3 := \frac{\sin ky \cdot th}{t\_2}\\ \mathbf{if}\;t\_1 \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), kx \cdot kx, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.06:\\ \;\;\;\;\frac{\sin th \cdot ky}{t\_2}\\ \mathbf{elif}\;t\_1 \leq 0.96:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 (hypot (sin ky) (sin kx)))
        (t_3 (/ (* (sin ky) th) t_2)))
   (if (<= t_1 -0.9995)
     (*
      (/
       (sin ky)
       (/
        (sqrt
         (fma
          (- 1.0 (cos (* ky 2.0)))
          2.0
          (*
           (*
            (fma
             (fma 0.17777777777777778 (* kx kx) -1.3333333333333333)
             (* kx kx)
             4.0)
            kx)
           kx)))
        2.0))
      (sin th))
     (if (<= t_1 -0.05)
       t_3
       (if (<= t_1 0.06)
         (/ (* (sin th) ky) t_2)
         (if (<= t_1 0.96) 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 = hypot(sin(ky), sin(kx));
	double t_3 = (sin(ky) * th) / t_2;
	double tmp;
	if (t_1 <= -0.9995) {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((ky * 2.0))), 2.0, ((fma(fma(0.17777777777777778, (kx * kx), -1.3333333333333333), (kx * kx), 4.0) * kx) * kx))) / 2.0)) * sin(th);
	} else if (t_1 <= -0.05) {
		tmp = t_3;
	} else if (t_1 <= 0.06) {
		tmp = (sin(th) * ky) / t_2;
	} else if (t_1 <= 0.96) {
		tmp = t_3;
	} 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))))
	t_2 = hypot(sin(ky), sin(kx))
	t_3 = Float64(Float64(sin(ky) * th) / t_2)
	tmp = 0.0
	if (t_1 <= -0.9995)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(ky * 2.0))), 2.0, Float64(Float64(fma(fma(0.17777777777777778, Float64(kx * kx), -1.3333333333333333), Float64(kx * kx), 4.0) * kx) * kx))) / 2.0)) * sin(th));
	elseif (t_1 <= -0.05)
		tmp = t_3;
	elseif (t_1 <= 0.06)
		tmp = Float64(Float64(sin(th) * ky) / t_2);
	elseif (t_1 <= 0.96)
		tmp = t_3;
	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]}, Block[{t$95$2 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9995], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(0.17777777777777778 * N[(kx * kx), $MachinePrecision] + -1.3333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 4.0), $MachinePrecision] * kx), $MachinePrecision] * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$3, If[LessEqual[t$95$1, 0.06], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 0.96], 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))))) < -0.99950000000000006

   1. Initial program 88.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{{kx}^{2} \cdot \left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right)}\right)}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot {kx}^{2}}\right)}}{2}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(kx \cdot kx\right)}\right)}}{2}} \cdot \sin th \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right)} \cdot kx\right)}}{2}} \cdot \sin th \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\color{blue}{\left({kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right) + 4\right)} \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\left(\color{blue}{\left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right) \cdot {kx}^{2}} + 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\color{blue}{\mathsf{fma}\left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}, {kx}^{2}, 4\right)} \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      9. sub-neg N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\color{blue}{\frac{8}{45} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\frac{8}{45} \cdot {kx}^{2} + \color{blue}{\frac{-4}{3}}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{8}{45}, {kx}^{2}, \frac{-4}{3}\right)}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, \color{blue}{kx \cdot kx}, \frac{-4}{3}\right), {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, \color{blue}{kx \cdot kx}, \frac{-4}{3}\right), {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, kx \cdot kx, \frac{-4}{3}\right), \color{blue}{kx \cdot kx}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      15. lower-*.f64 60.2

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), \color{blue}{kx \cdot kx}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), kx \cdot kx, 4\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

   if -0.99950000000000006 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 0.059999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   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. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 46.3

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

   7. Applied rewrites 46.3%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.059999999999999998

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 97.3

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

   7. Applied rewrites 97.3%

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

   if 0.95999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.4

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

   5. Applied rewrites 92.4%

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

Alternative 7: 68.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;t\_1 \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), kx \cdot kx, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.001:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.96:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (/ (* (sin ky) th) (hypot (sin ky) (sin kx)))))
   (if (<= t_1 -0.9995)
     (*
      (/
       (sin ky)
       (/
        (sqrt
         (fma
          (- 1.0 (cos (* ky 2.0)))
          2.0
          (*
           (*
            (fma
             (fma 0.17777777777777778 (* kx kx) -1.3333333333333333)
             (* kx kx)
             4.0)
            kx)
           kx)))
        2.0))
      (sin th))
     (if (<= t_1 -0.2)
       t_2
       (if (<= t_1 0.001)
         (*
          (/
           (* (fma -0.16666666666666666 (* ky ky) 1.0) ky)
           (/
            (sqrt
             (fma
              (- 1.0 (cos (* 2.0 kx)))
              2.0
              (*
               (*
                (fma
                 (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
                 (* ky ky)
                 4.0)
                ky)
               ky)))
            2.0))
          (sin th))
         (if (<= t_1 0.96) t_2 (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 = (sin(ky) * th) / hypot(sin(ky), sin(kx));
	double tmp;
	if (t_1 <= -0.9995) {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((ky * 2.0))), 2.0, ((fma(fma(0.17777777777777778, (kx * kx), -1.3333333333333333), (kx * kx), 4.0) * kx) * kx))) / 2.0)) * sin(th);
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 0.001) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * ky) / (sqrt(fma((1.0 - cos((2.0 * kx))), 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else if (t_1 <= 0.96) {
		tmp = t_2;
	} 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))))
	t_2 = Float64(Float64(sin(ky) * th) / hypot(sin(ky), sin(kx)))
	tmp = 0.0
	if (t_1 <= -0.9995)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(ky * 2.0))), 2.0, Float64(Float64(fma(fma(0.17777777777777778, Float64(kx * kx), -1.3333333333333333), Float64(kx * kx), 4.0) * kx) * kx))) / 2.0)) * sin(th));
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 0.001)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * kx))), 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	elseif (t_1 <= 0.96)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9995], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(0.17777777777777778 * N[(kx * kx), $MachinePrecision] + -1.3333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 4.0), $MachinePrecision] * kx), $MachinePrecision] * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 0.001], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.96], t$95$2, N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99950000000000006

   1. Initial program 88.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{{kx}^{2} \cdot \left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right)}\right)}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot {kx}^{2}}\right)}}{2}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(kx \cdot kx\right)}\right)}}{2}} \cdot \sin th \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right)} \cdot kx\right)}}{2}} \cdot \sin th \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\color{blue}{\left({kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right) + 4\right)} \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\left(\color{blue}{\left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right) \cdot {kx}^{2}} + 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\color{blue}{\mathsf{fma}\left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}, {kx}^{2}, 4\right)} \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      9. sub-neg N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\color{blue}{\frac{8}{45} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\frac{8}{45} \cdot {kx}^{2} + \color{blue}{\frac{-4}{3}}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{8}{45}, {kx}^{2}, \frac{-4}{3}\right)}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, \color{blue}{kx \cdot kx}, \frac{-4}{3}\right), {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, \color{blue}{kx \cdot kx}, \frac{-4}{3}\right), {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, kx \cdot kx, \frac{-4}{3}\right), \color{blue}{kx \cdot kx}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      15. lower-*.f64 60.2

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), \color{blue}{kx \cdot kx}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), kx \cdot kx, 4\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

   if -0.99950000000000006 < (/.f64 (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 1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   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. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 46.3

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

   7. Applied rewrites 46.3%

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

   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))))) < 1e-3

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 75.2%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. lower-*.f64 75.2

         \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 75.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   if 0.95999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.4

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

   5. Applied rewrites 92.4%

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

Alternative 8: 68.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(2 \cdot kx\right)\\ t_2 := 1 - t\_1\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_4 := 1 - \cos \left(2 \cdot ky\right)\\ \mathbf{if}\;t\_3 \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), kx \cdot kx, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{{\left(2 \cdot \left(t\_4 + t\_2\right)\right)}^{-1}}\\ \mathbf{elif}\;t\_3 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(t\_2, 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.96:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(t\_1 - t\_4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* 2.0 kx)))
        (t_2 (- 1.0 t_1))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_4 (- 1.0 (cos (* 2.0 ky)))))
   (if (<= t_3 -0.9995)
     (*
      (/
       (sin ky)
       (/
        (sqrt
         (fma
          (- 1.0 (cos (* ky 2.0)))
          2.0
          (*
           (*
            (fma
             (fma 0.17777777777777778 (* kx kx) -1.3333333333333333)
             (* kx kx)
             4.0)
            kx)
           kx)))
        2.0))
      (sin th))
     (if (<= t_3 -0.2)
       (* (* th (* 2.0 (sin ky))) (sqrt (pow (* 2.0 (+ t_4 t_2)) -1.0)))
       (if (<= t_3 0.02)
         (*
          (/
           (* (fma -0.16666666666666666 (* ky ky) 1.0) ky)
           (/
            (sqrt
             (fma
              t_2
              2.0
              (*
               (*
                (fma
                 (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
                 (* ky ky)
                 4.0)
                ky)
               ky)))
            2.0))
          (sin th))
         (if (<= t_3 0.96)
           (* (* 2.0 (* (sin ky) th)) (sqrt (/ 0.5 (- 1.0 (- t_1 t_4)))))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((2.0 * kx));
	double t_2 = 1.0 - t_1;
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_4 = 1.0 - cos((2.0 * ky));
	double tmp;
	if (t_3 <= -0.9995) {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((ky * 2.0))), 2.0, ((fma(fma(0.17777777777777778, (kx * kx), -1.3333333333333333), (kx * kx), 4.0) * kx) * kx))) / 2.0)) * sin(th);
	} else if (t_3 <= -0.2) {
		tmp = (th * (2.0 * sin(ky))) * sqrt(pow((2.0 * (t_4 + t_2)), -1.0));
	} else if (t_3 <= 0.02) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * ky) / (sqrt(fma(t_2, 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else if (t_3 <= 0.96) {
		tmp = (2.0 * (sin(ky) * th)) * sqrt((0.5 / (1.0 - (t_1 - t_4))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(2.0 * kx))
	t_2 = Float64(1.0 - t_1)
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_4 = Float64(1.0 - cos(Float64(2.0 * ky)))
	tmp = 0.0
	if (t_3 <= -0.9995)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(ky * 2.0))), 2.0, Float64(Float64(fma(fma(0.17777777777777778, Float64(kx * kx), -1.3333333333333333), Float64(kx * kx), 4.0) * kx) * kx))) / 2.0)) * sin(th));
	elseif (t_3 <= -0.2)
		tmp = Float64(Float64(th * Float64(2.0 * sin(ky))) * sqrt((Float64(2.0 * Float64(t_4 + t_2)) ^ -1.0)));
	elseif (t_3 <= 0.02)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(t_2, 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	elseif (t_3 <= 0.96)
		tmp = Float64(Float64(2.0 * Float64(sin(ky) * th)) * sqrt(Float64(0.5 / Float64(1.0 - Float64(t_1 - t_4)))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = 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$4 = N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.9995], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(0.17777777777777778 * N[(kx * kx), $MachinePrecision] + -1.3333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 4.0), $MachinePrecision] * kx), $MachinePrecision] * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], N[(N[(th * N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(2.0 * N[(t$95$4 + t$95$2), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.02], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(t$95$2 * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.96], N[(N[(2.0 * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(t$95$1 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99950000000000006

   1. Initial program 88.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{{kx}^{2} \cdot \left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right)}\right)}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot {kx}^{2}}\right)}}{2}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(kx \cdot kx\right)}\right)}}{2}} \cdot \sin th \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + {kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right)\right) \cdot kx\right)} \cdot kx\right)}}{2}} \cdot \sin th \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\color{blue}{\left({kx}^{2} \cdot \left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right) + 4\right)} \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\left(\color{blue}{\left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}\right) \cdot {kx}^{2}} + 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\color{blue}{\mathsf{fma}\left(\frac{8}{45} \cdot {kx}^{2} - \frac{4}{3}, {kx}^{2}, 4\right)} \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      9. sub-neg N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\color{blue}{\frac{8}{45} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\frac{8}{45} \cdot {kx}^{2} + \color{blue}{\frac{-4}{3}}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{8}{45}, {kx}^{2}, \frac{-4}{3}\right)}, {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      12. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, \color{blue}{kx \cdot kx}, \frac{-4}{3}\right), {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, \color{blue}{kx \cdot kx}, \frac{-4}{3}\right), {kx}^{2}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      14. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{8}{45}, kx \cdot kx, \frac{-4}{3}\right), \color{blue}{kx \cdot kx}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      15. lower-*.f64 60.2

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), \color{blue}{kx \cdot kx}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), kx \cdot kx, 4\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

   if -0.99950000000000006 < (/.f64 (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 99.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. *-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

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

   7. Applied rewrites 51.8%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 74.4%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. lower-*.f64 74.5

         \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 74.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 6.3%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   10. Applied rewrites 38.9%

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

   if 0.95999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(0.17777777777777778, kx \cdot kx, -1.3333333333333333\right), kx \cdot kx, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{{\left(2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.96:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := 1 - \cos \left(2 \cdot ky\right)\\ t_3 := \cos \left(2 \cdot kx\right)\\ t_4 := 1 - t\_3\\ \mathbf{if}\;t\_1 \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(-1.3333333333333333, kx \cdot kx, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{{\left(2 \cdot \left(t\_2 + t\_4\right)\right)}^{-1}}\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(t\_4, 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.96:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(t\_3 - t\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 (cos (* 2.0 kx)))
        (t_4 (- 1.0 t_3)))
   (if (<= t_1 -0.9995)
     (*
      (/
       (sin ky)
       (/
        (sqrt
         (fma
          (- 1.0 (cos (* ky 2.0)))
          2.0
          (* (* (fma -1.3333333333333333 (* kx kx) 4.0) kx) kx)))
        2.0))
      (sin th))
     (if (<= t_1 -0.2)
       (* (* th (* 2.0 (sin ky))) (sqrt (pow (* 2.0 (+ t_2 t_4)) -1.0)))
       (if (<= t_1 0.02)
         (*
          (/
           (* (fma -0.16666666666666666 (* ky ky) 1.0) ky)
           (/
            (sqrt
             (fma
              t_4
              2.0
              (*
               (*
                (fma
                 (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
                 (* ky ky)
                 4.0)
                ky)
               ky)))
            2.0))
          (sin th))
         (if (<= t_1 0.96)
           (* (* 2.0 (* (sin ky) th)) (sqrt (/ 0.5 (- 1.0 (- t_3 t_2)))))
           (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 = cos((2.0 * kx));
	double t_4 = 1.0 - t_3;
	double tmp;
	if (t_1 <= -0.9995) {
		tmp = (sin(ky) / (sqrt(fma((1.0 - cos((ky * 2.0))), 2.0, ((fma(-1.3333333333333333, (kx * kx), 4.0) * kx) * kx))) / 2.0)) * sin(th);
	} else if (t_1 <= -0.2) {
		tmp = (th * (2.0 * sin(ky))) * sqrt(pow((2.0 * (t_2 + t_4)), -1.0));
	} else if (t_1 <= 0.02) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * ky) / (sqrt(fma(t_4, 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else if (t_1 <= 0.96) {
		tmp = (2.0 * (sin(ky) * th)) * sqrt((0.5 / (1.0 - (t_3 - t_2))));
	} 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))))
	t_2 = Float64(1.0 - cos(Float64(2.0 * ky)))
	t_3 = cos(Float64(2.0 * kx))
	t_4 = Float64(1.0 - t_3)
	tmp = 0.0
	if (t_1 <= -0.9995)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(ky * 2.0))), 2.0, Float64(Float64(fma(-1.3333333333333333, Float64(kx * kx), 4.0) * kx) * kx))) / 2.0)) * sin(th));
	elseif (t_1 <= -0.2)
		tmp = Float64(Float64(th * Float64(2.0 * sin(ky))) * sqrt((Float64(2.0 * Float64(t_2 + t_4)) ^ -1.0)));
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(t_4, 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	elseif (t_1 <= 0.96)
		tmp = Float64(Float64(2.0 * Float64(sin(ky) * th)) * sqrt(Float64(0.5 / Float64(1.0 - Float64(t_3 - t_2)))));
	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]}, Block[{t$95$2 = N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - t$95$3), $MachinePrecision]}, If[LessEqual[t$95$1, -0.9995], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(-1.3333333333333333 * N[(kx * kx), $MachinePrecision] + 4.0), $MachinePrecision] * kx), $MachinePrecision] * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], N[(N[(th * N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(2.0 * N[(t$95$2 + t$95$4), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(t$95$4 * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.96], N[(N[(2.0 * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(t$95$3 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.99950000000000006

   1. Initial program 88.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{{kx}^{2} \cdot \left(4 + \frac{-4}{3} \cdot {kx}^{2}\right)}\right)}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(4 + \frac{-4}{3} \cdot {kx}^{2}\right) \cdot {kx}^{2}}\right)}}{2}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(4 + \frac{-4}{3} \cdot {kx}^{2}\right) \cdot \color{blue}{\left(kx \cdot kx\right)}\right)}}{2}} \cdot \sin th \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + \frac{-4}{3} \cdot {kx}^{2}\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + \frac{-4}{3} \cdot {kx}^{2}\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\left(4 + \frac{-4}{3} \cdot {kx}^{2}\right) \cdot kx\right)} \cdot kx\right)}}{2}} \cdot \sin th \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\color{blue}{\left(\frac{-4}{3} \cdot {kx}^{2} + 4\right)} \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\color{blue}{\mathsf{fma}\left(\frac{-4}{3}, {kx}^{2}, 4\right)} \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(\frac{-4}{3}, \color{blue}{kx \cdot kx}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 60.2

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(-1.3333333333333333, \color{blue}{kx \cdot kx}, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \color{blue}{\left(\mathsf{fma}\left(-1.3333333333333333, kx \cdot kx, 4\right) \cdot kx\right) \cdot kx}\right)}}{2}} \cdot \sin th \]

   if -0.99950000000000006 < (/.f64 (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 99.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. *-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

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

   7. Applied rewrites 51.8%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 74.4%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. lower-*.f64 74.5

         \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 74.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 6.3%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   10. Applied rewrites 38.9%

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

   if 0.95999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.9995:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(\mathsf{fma}\left(-1.3333333333333333, kx \cdot kx, 4\right) \cdot kx\right) \cdot kx\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{{\left(2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.96:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(2 \cdot kx\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := 1 - \cos \left(2 \cdot ky\right)\\ t_4 := 1 - t\_1\\ \mathbf{if}\;t\_2 \leq -0.998:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{t\_3 \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{{\left(2 \cdot \left(t\_3 + t\_4\right)\right)}^{-1}}\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(t\_4, 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.96:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(t\_1 - t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* 2.0 kx)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (- 1.0 (cos (* 2.0 ky))))
        (t_4 (- 1.0 t_1)))
   (if (<= t_2 -0.998)
     (* (/ (sin ky) (/ (sqrt (* t_3 2.0)) 2.0)) (sin th))
     (if (<= t_2 -0.2)
       (* (* th (* 2.0 (sin ky))) (sqrt (pow (* 2.0 (+ t_3 t_4)) -1.0)))
       (if (<= t_2 0.02)
         (*
          (/
           (* (fma -0.16666666666666666 (* ky ky) 1.0) ky)
           (/
            (sqrt
             (fma
              t_4
              2.0
              (*
               (*
                (fma
                 (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
                 (* ky ky)
                 4.0)
                ky)
               ky)))
            2.0))
          (sin th))
         (if (<= t_2 0.96)
           (* (* 2.0 (* (sin ky) th)) (sqrt (/ 0.5 (- 1.0 (- t_1 t_3)))))
           (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((2.0 * kx));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = 1.0 - cos((2.0 * ky));
	double t_4 = 1.0 - t_1;
	double tmp;
	if (t_2 <= -0.998) {
		tmp = (sin(ky) / (sqrt((t_3 * 2.0)) / 2.0)) * sin(th);
	} else if (t_2 <= -0.2) {
		tmp = (th * (2.0 * sin(ky))) * sqrt(pow((2.0 * (t_3 + t_4)), -1.0));
	} else if (t_2 <= 0.02) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * ky) / (sqrt(fma(t_4, 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else if (t_2 <= 0.96) {
		tmp = (2.0 * (sin(ky) * th)) * sqrt((0.5 / (1.0 - (t_1 - t_3))));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(2.0 * kx))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(1.0 - cos(Float64(2.0 * ky)))
	t_4 = Float64(1.0 - t_1)
	tmp = 0.0
	if (t_2 <= -0.998)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(t_3 * 2.0)) / 2.0)) * sin(th));
	elseif (t_2 <= -0.2)
		tmp = Float64(Float64(th * Float64(2.0 * sin(ky))) * sqrt((Float64(2.0 * Float64(t_3 + t_4)) ^ -1.0)));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(t_4, 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	elseif (t_2 <= 0.96)
		tmp = Float64(Float64(2.0 * Float64(sin(ky) * th)) * sqrt(Float64(0.5 / Float64(1.0 - Float64(t_1 - t_3)))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -0.998], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(t$95$3 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], N[(N[(th * N[(2.0 * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(2.0 * N[(t$95$3 + t$95$4), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(t$95$4 * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.96], N[(N[(2.0 * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(t$95$1 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 88.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

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

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

      1. *-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

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

   7. Applied rewrites 51.8%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 74.4%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. lower-*.f64 74.5

         \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 74.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 6.3%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   10. Applied rewrites 38.9%

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

   if 0.95999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.4

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

   5. Applied rewrites 92.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.998:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\left(th \cdot \left(2 \cdot \sin ky\right)\right) \cdot \sqrt{{\left(2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)\right)}^{-1}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.96:\\ \;\;\;\;\left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(2 \cdot kx\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := 1 - \cos \left(2 \cdot ky\right)\\ t_4 := \left(2 \cdot \left(\sin ky \cdot th\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(t\_1 - t\_3\right)}}\\ \mathbf{if}\;t\_2 \leq -0.998:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{t\_3 \cdot 2}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - t\_1, 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.96:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (cos (* 2.0 kx)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (- 1.0 (cos (* 2.0 ky))))
        (t_4 (* (* 2.0 (* (sin ky) th)) (sqrt (/ 0.5 (- 1.0 (- t_1 t_3)))))))
   (if (<= t_2 -0.998)
     (* (/ (sin ky) (/ (sqrt (* t_3 2.0)) 2.0)) (sin th))
     (if (<= t_2 -0.2)
       t_4
       (if (<= t_2 0.02)
         (*
          (/
           (* (fma -0.16666666666666666 (* ky ky) 1.0) ky)
           (/
            (sqrt
             (fma
              (- 1.0 t_1)
              2.0
              (*
               (*
                (fma
                 (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
                 (* ky ky)
                 4.0)
                ky)
               ky)))
            2.0))
          (sin th))
         (if (<= t_2 0.96) t_4 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = cos((2.0 * kx));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = 1.0 - cos((2.0 * ky));
	double t_4 = (2.0 * (sin(ky) * th)) * sqrt((0.5 / (1.0 - (t_1 - t_3))));
	double tmp;
	if (t_2 <= -0.998) {
		tmp = (sin(ky) / (sqrt((t_3 * 2.0)) / 2.0)) * sin(th);
	} else if (t_2 <= -0.2) {
		tmp = t_4;
	} else if (t_2 <= 0.02) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * ky) / (sqrt(fma((1.0 - t_1), 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else if (t_2 <= 0.96) {
		tmp = t_4;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = cos(Float64(2.0 * kx))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(1.0 - cos(Float64(2.0 * ky)))
	t_4 = Float64(Float64(2.0 * Float64(sin(ky) * th)) * sqrt(Float64(0.5 / Float64(1.0 - Float64(t_1 - t_3)))))
	tmp = 0.0
	if (t_2 <= -0.998)
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(t_3 * 2.0)) / 2.0)) * sin(th));
	elseif (t_2 <= -0.2)
		tmp = t_4;
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(Float64(1.0 - t_1), 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	elseif (t_2 <= 0.96)
		tmp = t_4;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 * N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 / N[(1.0 - N[(t$95$1 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.998], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(t$95$3 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.2], t$95$4, If[LessEqual[t$95$2, 0.02], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.96], t$95$4, 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))))) < -0.998

   1. Initial program 88.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   if -0.998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.95999999999999996

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 6.0%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in th around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. associate-+l- N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

   10. Applied rewrites 45.4%

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

   if -0.20000000000000001 < (/.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))))) < 0.0200000000000000004

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 74.4%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. lower-*.f64 74.5

         \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 74.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   if 0.95999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 92.4

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

   5. Applied rewrites 92.4%

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

Alternative 12: 54.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;{\left({\sin th}^{2}\right)}^{0.5}\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.05)
     (pow (pow (sin th) 2.0) 0.5)
     (if (<= t_1 0.2)
       (*
        (/
         (*
          (fma
           (fma 0.008333333333333333 (* ky ky) -0.16666666666666666)
           (* ky ky)
           1.0)
          ky)
         (/
          (sqrt
           (fma
            (- 1.0 (cos (* 2.0 kx)))
            2.0
            (*
             (*
              (fma
               (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
               (* ky ky)
               4.0)
              ky)
             ky)))
          2.0))
        (sin th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = pow(pow(sin(th), 2.0), 0.5);
	} else if (t_1 <= 0.2) {
		tmp = ((fma(fma(0.008333333333333333, (ky * ky), -0.16666666666666666), (ky * ky), 1.0) * ky) / (sqrt(fma((1.0 - cos((2.0 * kx))), 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} 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 <= -0.05)
		tmp = (sin(th) ^ 2.0) ^ 0.5;
	elseif (t_1 <= 0.2)
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666), Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * kx))), 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	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, -0.05], N[Power[N[Power[N[Sin[th], $MachinePrecision], 2.0], $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$1, 0.2], N[(N[(N[(N[(N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 93.9%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 1.6%

      \[\leadsto e^{\log \left({\sin th}^{-1}\right) \cdot -1} \]
   8. Step-by-step derivation

   9. Applied rewrites 20.4%

      \[\leadsto {\left({\sin th}^{2}\right)}^{\color{blue}{0.5}} \]

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.20000000000000001

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 74.5%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) \cdot {ky}^{2}} + 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, ky \cdot ky, \frac{-1}{6}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      12. lower-*.f64 74.5

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 74.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

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

   1. Initial program 96.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.1

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

   5. Applied rewrites 62.1%

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

Alternative 13: 54.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.2:\\ \;\;\;\;{\left({\sin th}^{2}\right)}^{0.5}\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.2)
     (pow (pow (sin th) 2.0) 0.5)
     (if (<= t_1 0.2)
       (*
        (/
         (* (fma -0.16666666666666666 (* ky ky) 1.0) ky)
         (/
          (sqrt
           (fma
            (- 1.0 (cos (* 2.0 kx)))
            2.0
            (*
             (*
              (fma
               (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
               (* ky ky)
               4.0)
              ky)
             ky)))
          2.0))
        (sin th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = pow(pow(sin(th), 2.0), 0.5);
	} else if (t_1 <= 0.2) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * ky) / (sqrt(fma((1.0 - cos((2.0 * kx))), 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} 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 <= -0.2)
		tmp = (sin(th) ^ 2.0) ^ 0.5;
	elseif (t_1 <= 0.2)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * kx))), 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	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, -0.2], N[Power[N[Power[N[Sin[th], $MachinePrecision], 2.0], $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t$95$1, 0.2], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 93.8%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 3.0

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 1.6%

      \[\leadsto e^{\log \left({\sin th}^{-1}\right) \cdot -1} \]
   8. Step-by-step derivation

   9. Applied rewrites 20.8%

      \[\leadsto {\left({\sin th}^{2}\right)}^{\color{blue}{0.5}} \]

   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))))) < 0.20000000000000001

   1. Initial program 99.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 73.0%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. lower-*.f64 72.7

         \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 72.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

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

   1. Initial program 96.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.1

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

   5. Applied rewrites 62.1%

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

Alternative 14: 46.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-8}:\\ \;\;\;\;\left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{{\left(1 - \cos \left(2 \cdot kx\right)\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-8)
   (*
    (* 2.0 (* (* (sin th) ky) (sqrt 0.5)))
    (sqrt (pow (- 1.0 (cos (* 2.0 kx))) -1.0)))
   (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)))) <= 1e-8) {
		tmp = (2.0 * ((sin(th) * ky) * sqrt(0.5))) * sqrt(pow((1.0 - cos((2.0 * kx))), -1.0));
	} 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)))) <= 1d-8) then
        tmp = (2.0d0 * ((sin(th) * ky) * sqrt(0.5d0))) * sqrt(((1.0d0 - cos((2.0d0 * kx))) ** (-1.0d0)))
    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)))) <= 1e-8) {
		tmp = (2.0 * ((Math.sin(th) * ky) * Math.sqrt(0.5))) * Math.sqrt(Math.pow((1.0 - Math.cos((2.0 * kx))), -1.0));
	} 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)))) <= 1e-8:
		tmp = (2.0 * ((math.sin(th) * ky) * math.sqrt(0.5))) * math.sqrt(math.pow((1.0 - math.cos((2.0 * kx))), -1.0))
	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)))) <= 1e-8)
		tmp = Float64(Float64(2.0 * Float64(Float64(sin(th) * ky) * sqrt(0.5))) * sqrt((Float64(1.0 - cos(Float64(2.0 * kx))) ^ -1.0)));
	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)))) <= 1e-8)
		tmp = (2.0 * ((sin(th) * ky) * sqrt(0.5))) * sqrt(((1.0 - cos((2.0 * kx))) ^ -1.0));
	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], 1e-8], N[(N[(2.0 * N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $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))))) < 1e-8

   1. Initial program 96.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 78.2%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 38.5

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

   10. Applied rewrites 38.5%

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

   if 1e-8 < (/.f64 (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 96.4%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.5

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

   5. Applied rewrites 59.5%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-8}:\\ \;\;\;\;\left(2 \cdot \left(\left(\sin th \cdot ky\right) \cdot \sqrt{0.5}\right)\right) \cdot \sqrt{{\left(1 - \cos \left(2 \cdot kx\right)\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.2)
   (*
    (/
     (* (fma -0.16666666666666666 (* ky ky) 1.0) ky)
     (/
      (sqrt
       (fma
        (- 1.0 (cos (* 2.0 kx)))
        2.0
        (*
         (*
          (fma
           (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
           (* ky ky)
           4.0)
          ky)
         ky)))
      2.0))
    (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.2) {
		tmp = ((fma(-0.16666666666666666, (ky * ky), 1.0) * ky) / (sqrt(fma((1.0 - cos((2.0 * kx))), 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else {
		tmp = 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)))) <= 0.2)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * kx))), 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.2], N[(N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-div N/A

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 46.7%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. lower-*.f64 45.8

         \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 45.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

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

   1. Initial program 96.3%

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

   3. Taylor expanded in kx around 0

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

      1. lower-sin.f64 62.1

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

   5. Applied rewrites 62.1%

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

Alternative 16: 43.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.02)
   (* (/ (* (fma (* ky ky) -0.16666666666666666 1.0) ky) (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.02) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / sin(kx)) * sin(th);
	} else {
		tmp = 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)))) <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 35.4

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

   5. Applied rewrites 35.4%

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

   6. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{-1}{6}, 1\right) \cdot ky}{\sin kx} \cdot \sin th \]

      7. lower-*.f64 33.1

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, -0.16666666666666666, 1\right) \cdot ky}{\sin kx} \cdot \sin th \]

   8. Applied rewrites 33.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\sin kx} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 96.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 60.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 43.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.02)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.02) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.02d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.02) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.02:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 96.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 33.2

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 33.2%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 96.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 60.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 43.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.02)
   (/ (* (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)))) <= 0.02) {
		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)))) <= 0.02d0) 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)))) <= 0.02) {
		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)))) <= 0.02:
		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)))) <= 0.02)
		tmp = Float64(Float64(sin(th) * 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)))) <= 0.02)
		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], 0.02], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $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))))) < 0.0200000000000000004

   1. Initial program 96.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      4. lower-/.f64 96.6

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \cdot \sin th \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \cdot \sin th \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}}{\sin ky}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \cdot \sin th \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}}{\sin ky}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \cdot \sin th \]

      12. lower-hypot.f64 99.5

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \cdot \sin th \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sin kx} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot ky}}{\sin kx} \]

      4. lower-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin th} \cdot ky}{\sin kx} \]

      5. lower-sin.f64 33.3

         \[\leadsto \frac{\sin th \cdot ky}{\color{blue}{\sin kx}} \]

   7. Applied rewrites 33.3%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   if 0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 96.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 60.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 3.6 \cdot 10^{-66}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 3.6e-66)
   (* (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)))) <= 3.6e-66) {
		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)))) <= 3.6d-66) 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)))) <= 3.6e-66) {
		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)))) <= 3.6e-66:
		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)))) <= 3.6e-66)
		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)))) <= 3.6e-66)
		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], 3.6e-66], 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))))) < 3.60000000000000012e-66

   1. Initial program 96.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 3.6%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.6%

      \[\leadsto \mathsf{fma}\left({th}^{3}, \color{blue}{-0.16666666666666666}, th\right) \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 13.7%

       \[\leadsto {th}^{3} \cdot -0.16666666666666666 \]

   if 3.60000000000000012e-66 < (/.f64 (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 96.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 55.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 46.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-191}:\\ \;\;\;\;{\left(\frac{\frac{\sin kx}{\sin th}}{\sin ky}\right)}^{-1}\\ \mathbf{elif}\;\sin ky \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-191)
   (pow (/ (/ (sin kx) (sin th)) (sin ky)) -1.0)
   (if (<= (sin ky) 0.01)
     (*
      (/
       (*
        (fma
         (fma 0.008333333333333333 (* ky ky) -0.16666666666666666)
         (* ky ky)
         1.0)
        ky)
       (/
        (sqrt
         (fma
          (- 1.0 (cos (* 2.0 kx)))
          2.0
          (*
           (*
            (fma
             (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
             (* ky ky)
             4.0)
            ky)
           ky)))
        2.0))
      (sin th))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-191) {
		tmp = pow(((sin(kx) / sin(th)) / sin(ky)), -1.0);
	} else if (sin(ky) <= 0.01) {
		tmp = ((fma(fma(0.008333333333333333, (ky * ky), -0.16666666666666666), (ky * ky), 1.0) * ky) / (sqrt(fma((1.0 - cos((2.0 * kx))), 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-191)
		tmp = Float64(Float64(sin(kx) / sin(th)) / sin(ky)) ^ -1.0;
	elseif (sin(ky) <= 0.01)
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666), Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * kx))), 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-191], N[Power[N[(N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.01], N[(N[(N[(N[(N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < 2e-191

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 30.3

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 30.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sin kx} \]

      6. lower-*.f64 30.3

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sin kx} \]

   7. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sin kx}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sin kx}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin kx}{\sin th \cdot \sin ky}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin kx}{\sin th \cdot \sin ky}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sin kx}{\color{blue}{\sin th \cdot \sin ky}}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}} \]

      7. lower-/.f64 29.9

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\sin kx}{\sin th}}}{\sin ky}} \]

   9. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}} \]

   if 2e-191 < (sin.f64 ky) < 0.0100000000000000002

   1. Initial program 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 61.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 86.0%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) \cdot {ky}^{2}} + 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, ky \cdot ky, \frac{-1}{6}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      12. lower-*.f64 86.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 86.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   if 0.0100000000000000002 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 56.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-191}:\\ \;\;\;\;{\left(\frac{\frac{\sin kx}{\sin th}}{\sin ky}\right)}^{-1}\\ \mathbf{elif}\;\sin ky \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 45.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-191}:\\ \;\;\;\;{\left(\frac{\sin kx}{\sin th \cdot \sin ky}\right)}^{-1}\\ \mathbf{elif}\;\sin ky \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-191)
   (pow (/ (sin kx) (* (sin th) (sin ky))) -1.0)
   (if (<= (sin ky) 0.01)
     (*
      (/
       (*
        (fma
         (fma 0.008333333333333333 (* ky ky) -0.16666666666666666)
         (* ky ky)
         1.0)
        ky)
       (/
        (sqrt
         (fma
          (- 1.0 (cos (* 2.0 kx)))
          2.0
          (*
           (*
            (fma
             (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
             (* ky ky)
             4.0)
            ky)
           ky)))
        2.0))
      (sin th))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-191) {
		tmp = pow((sin(kx) / (sin(th) * sin(ky))), -1.0);
	} else if (sin(ky) <= 0.01) {
		tmp = ((fma(fma(0.008333333333333333, (ky * ky), -0.16666666666666666), (ky * ky), 1.0) * ky) / (sqrt(fma((1.0 - cos((2.0 * kx))), 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-191)
		tmp = Float64(sin(kx) / Float64(sin(th) * sin(ky))) ^ -1.0;
	elseif (sin(ky) <= 0.01)
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666), Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * kx))), 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-191], N[Power[N[(N[Sin[kx], $MachinePrecision] / N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.01], N[(N[(N[(N[(N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < 2e-191

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 30.3

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 30.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin kx}{\sin ky \cdot \sin th}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin kx}{\sin ky \cdot \sin th}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\sin kx}{\sin ky \cdot \sin th}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\sin kx}{\color{blue}{\sin th \cdot \sin ky}}} \]

      8. lower-*.f64 30.0

         \[\leadsto \frac{1}{\frac{\sin kx}{\color{blue}{\sin th \cdot \sin ky}}} \]

   7. Applied rewrites 30.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin kx}{\sin th \cdot \sin ky}}} \]

   if 2e-191 < (sin.f64 ky) < 0.0100000000000000002

   1. Initial program 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 61.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 86.0%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) \cdot {ky}^{2}} + 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, ky \cdot ky, \frac{-1}{6}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      12. lower-*.f64 86.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 86.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   if 0.0100000000000000002 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 56.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-191}:\\ \;\;\;\;{\left(\frac{\sin kx}{\sin th \cdot \sin ky}\right)}^{-1}\\ \mathbf{elif}\;\sin ky \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 45.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-191)
   (/ (* (sin th) (sin ky)) (sin kx))
   (if (<= (sin ky) 0.01)
     (*
      (/
       (*
        (fma
         (fma 0.008333333333333333 (* ky ky) -0.16666666666666666)
         (* ky ky)
         1.0)
        ky)
       (/
        (sqrt
         (fma
          (- 1.0 (cos (* 2.0 kx)))
          2.0
          (*
           (*
            (fma
             (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
             (* ky ky)
             4.0)
            ky)
           ky)))
        2.0))
      (sin th))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-191) {
		tmp = (sin(th) * sin(ky)) / sin(kx);
	} else if (sin(ky) <= 0.01) {
		tmp = ((fma(fma(0.008333333333333333, (ky * ky), -0.16666666666666666), (ky * ky), 1.0) * ky) / (sqrt(fma((1.0 - cos((2.0 * kx))), 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-191)
		tmp = Float64(Float64(sin(th) * sin(ky)) / sin(kx));
	elseif (sin(ky) <= 0.01)
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666), Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * kx))), 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-191], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.01], N[(N[(N[(N[(N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < 2e-191

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 30.3

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 30.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sin kx} \]

      6. lower-*.f64 30.3

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sin kx} \]

   7. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sin kx}} \]

   if 2e-191 < (sin.f64 ky) < 0.0100000000000000002

   1. Initial program 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 61.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 86.0%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) \cdot {ky}^{2}} + 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, ky \cdot ky, \frac{-1}{6}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      12. lower-*.f64 86.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 86.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   if 0.0100000000000000002 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 56.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 46.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\sin ky \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 2e-191)
   (* (/ (sin ky) (sin kx)) (sin th))
   (if (<= (sin ky) 0.01)
     (*
      (/
       (*
        (fma
         (fma 0.008333333333333333 (* ky ky) -0.16666666666666666)
         (* ky ky)
         1.0)
        ky)
       (/
        (sqrt
         (fma
          (- 1.0 (cos (* 2.0 kx)))
          2.0
          (*
           (*
            (fma
             (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
             (* ky ky)
             4.0)
            ky)
           ky)))
        2.0))
      (sin th))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 2e-191) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (sin(ky) <= 0.01) {
		tmp = ((fma(fma(0.008333333333333333, (ky * ky), -0.16666666666666666), (ky * ky), 1.0) * ky) / (sqrt(fma((1.0 - cos((2.0 * kx))), 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 2e-191)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (sin(ky) <= 0.01)
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666), Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * kx))), 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-191], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.01], N[(N[(N[(N[(N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < 2e-191

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 30.3

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 30.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 2e-191 < (sin.f64 ky) < 0.0100000000000000002

   1. Initial program 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 61.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 86.0%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) \cdot {ky}^{2}} + 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, ky \cdot ky, \frac{-1}{6}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      12. lower-*.f64 86.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 86.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   if 0.0100000000000000002 < (sin.f64 ky)

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 56.7

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky \end{array} \]

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(Float64(sin(th) / 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[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin ky} \]

   7. lower-/.f64 96.5

      \[\leadsto \color{blue}{\frac{\sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin ky \]

   10. +-commutative N/A

       \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin ky \]

   11. lift-pow.f64 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin ky \]

   12. unpow2 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin ky \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin ky \]

   14. unpow2 N/A

       \[\leadsto \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin ky \]

   15. lower-hypot.f64 99.5

       \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin ky \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
5. Add Preprocessing

Alternative 25: 75.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{-1}{ky} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot 2}{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.1e-5)
   (* (/ -1.0 (* (/ -1.0 ky) (hypot (sin kx) (sin ky)))) (sin th))
   (*
    (/
     (* (sin ky) 2.0)
     (sqrt (* 2.0 (+ (- 1.0 (cos (* 2.0 ky))) (- 1.0 (cos (* 2.0 kx)))))))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.1e-5) {
		tmp = (-1.0 / ((-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th);
	} else {
		tmp = ((sin(ky) * 2.0) / sqrt((2.0 * ((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx))))))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.1e-5) {
		tmp = (-1.0 / ((-1.0 / ky) * Math.hypot(Math.sin(kx), Math.sin(ky)))) * Math.sin(th);
	} else {
		tmp = ((Math.sin(ky) * 2.0) / Math.sqrt((2.0 * ((1.0 - Math.cos((2.0 * ky))) + (1.0 - Math.cos((2.0 * kx))))))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2.1e-5:
		tmp = (-1.0 / ((-1.0 / ky) * math.hypot(math.sin(kx), math.sin(ky)))) * math.sin(th)
	else:
		tmp = ((math.sin(ky) * 2.0) / math.sqrt((2.0 * ((1.0 - math.cos((2.0 * ky))) + (1.0 - math.cos((2.0 * kx))))))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.1e-5)
		tmp = Float64(Float64(-1.0 / Float64(Float64(-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th));
	else
		tmp = Float64(Float64(Float64(sin(ky) * 2.0) / sqrt(Float64(2.0 * Float64(Float64(1.0 - cos(Float64(2.0 * ky))) + Float64(1.0 - cos(Float64(2.0 * kx))))))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2.1e-5)
		tmp = (-1.0 / ((-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th);
	else
		tmp = ((sin(ky) * 2.0) / sqrt((2.0 * ((1.0 - cos((2.0 * ky))) + (1.0 - cos((2.0 * kx))))))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.1e-5], N[(N[(-1.0 / N[(N[(-1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[ky], $MachinePrecision] * 2.0), $MachinePrecision] / 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]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.09999999999999988e-5

   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      4. lower-/.f64 95.4

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \cdot \sin th \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \cdot \sin th \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}}{\sin ky}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \cdot \sin th \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}}{\sin ky}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \cdot \sin th \]

      12. lower-hypot.f64 99.6

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \cdot \sin th \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{\sin ky} \cdot \left(-\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{ky}} \cdot \left(-\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)} \cdot \sin th \]
   8. Step-by-step derivation

      1. lower-/.f64 60.6

         \[\leadsto \frac{1}{\color{blue}{\frac{-1}{ky}} \cdot \left(-\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)} \cdot \sin th \]

   9. Applied rewrites 60.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{ky}} \cdot \left(-\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)} \cdot \sin th \]

   if 2.09999999999999988e-5 < ky

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      4. lower-/.f64 99.6

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \cdot \sin th \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \cdot \sin th \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}}{\sin ky}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \cdot \sin th \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}}{\sin ky}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \cdot \sin th \]

      12. lower-hypot.f64 99.6

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \cdot \sin th \]

   6. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot 2}{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{-1}{ky} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot 2}{\sqrt{2 \cdot \left(\left(1 - \cos \left(2 \cdot ky\right)\right) + \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 75.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{-1}{ky} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.1e-5)
   (* (/ -1.0 (* (/ -1.0 ky) (hypot (sin kx) (sin ky)))) (sin th))
   (*
    (* 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 (ky <= 2.1e-5) {
		tmp = (-1.0 / ((-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th);
	} 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;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.1e-5) {
		tmp = (-1.0 / ((-1.0 / ky) * Math.hypot(Math.sin(kx), Math.sin(ky)))) * Math.sin(th);
	} else {
		tmp = (2.0 * (Math.sin(th) * Math.sin(ky))) * Math.sqrt((0.5 / (1.0 - (Math.cos((2.0 * kx)) - (1.0 - Math.cos((2.0 * ky)))))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2.1e-5:
		tmp = (-1.0 / ((-1.0 / ky) * math.hypot(math.sin(kx), math.sin(ky)))) * math.sin(th)
	else:
		tmp = (2.0 * (math.sin(th) * math.sin(ky))) * math.sqrt((0.5 / (1.0 - (math.cos((2.0 * kx)) - (1.0 - math.cos((2.0 * ky)))))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.1e-5)
		tmp = Float64(Float64(-1.0 / Float64(Float64(-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th));
	else
		tmp = Float64(Float64(2.0 * Float64(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

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2.1e-5)
		tmp = (-1.0 / ((-1.0 / ky) * hypot(sin(kx), sin(ky)))) * sin(th);
	else
		tmp = (2.0 * (sin(th) * sin(ky))) * sqrt((0.5 / (1.0 - (cos((2.0 * kx)) - (1.0 - cos((2.0 * ky)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.1e-5], N[(N[(-1.0 / N[(N[(-1.0 / ky), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Sin[th], $MachinePrecision] * 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]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.09999999999999988e-5

   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. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      4. lower-/.f64 95.4

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \cdot \sin th \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}{\sin ky}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{\sin ky}} \cdot \sin th \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}}{\sin ky}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{\sin ky}} \cdot \sin th \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}}{\sin ky}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{\sin ky}} \cdot \sin th \]

      12. lower-hypot.f64 99.6

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{\sin ky}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \cdot \sin th \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{\sin ky} \cdot \left(-\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{ky}} \cdot \left(-\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)} \cdot \sin th \]
   8. Step-by-step derivation

      1. lower-/.f64 60.6

         \[\leadsto \frac{1}{\color{blue}{\frac{-1}{ky}} \cdot \left(-\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)} \cdot \sin th \]

   9. Applied rewrites 60.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1}{ky}} \cdot \left(-\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)} \cdot \sin th \]

   if 2.09999999999999988e-5 < ky

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 5.6%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{2 \cdot \left(\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot \sin th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot \sin th\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(\sin ky \cdot \sin th\right)\right)} \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\sin th \cdot \sin ky\right)}\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      6. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sin th \cdot \color{blue}{\sin ky}\right)\right) \cdot \sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + 2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      9. distribute-lft-out N/A

         \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      10. associate-/r* N/A

          \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      11. metadata-eval N/A

          \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{\color{blue}{\frac{1}{2}}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

      13. associate-+l- N/A

          \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

      14. lower--.f64 N/A

          \[\leadsto \left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{\frac{1}{2}}{\color{blue}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]

   10. Applied rewrites 98.8%

       \[\leadsto \color{blue}{\left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{-1}{ky} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\sin th \cdot \sin ky\right)\right) \cdot \sqrt{\frac{0.5}{1 - \left(\cos \left(2 \cdot kx\right) - \left(1 - \cos \left(2 \cdot ky\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 46.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.4 \cdot 10^{-191}:\\ \;\;\;\;{\left(\frac{\frac{\sin kx}{\sin th}}{\sin ky}\right)}^{-1}\\ \mathbf{elif}\;ky \leq 0.68:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.4e-191)
   (pow (/ (/ (sin kx) (sin th)) (sin ky)) -1.0)
   (if (<= ky 0.68)
     (*
      (/
       (*
        (fma
         (fma 0.008333333333333333 (* ky ky) -0.16666666666666666)
         (* ky ky)
         1.0)
        ky)
       (/
        (sqrt
         (fma
          (- 1.0 (cos (* 2.0 kx)))
          2.0
          (*
           (*
            (fma
             (fma (* ky ky) 0.17777777777777778 -1.3333333333333333)
             (* ky ky)
             4.0)
            ky)
           ky)))
        2.0))
      (sin th))
     (*
      (/ (sin ky) (/ (sqrt (* (- 1.0 (cos (* 2.0 ky))) 2.0)) 2.0))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.4e-191) {
		tmp = pow(((sin(kx) / sin(th)) / sin(ky)), -1.0);
	} else if (ky <= 0.68) {
		tmp = ((fma(fma(0.008333333333333333, (ky * ky), -0.16666666666666666), (ky * ky), 1.0) * ky) / (sqrt(fma((1.0 - cos((2.0 * kx))), 2.0, ((fma(fma((ky * ky), 0.17777777777777778, -1.3333333333333333), (ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th);
	} else {
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * ky))) * 2.0)) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.4e-191)
		tmp = Float64(Float64(sin(kx) / sin(th)) / sin(ky)) ^ -1.0;
	elseif (ky <= 0.68)
		tmp = Float64(Float64(Float64(fma(fma(0.008333333333333333, Float64(ky * ky), -0.16666666666666666), Float64(ky * ky), 1.0) * ky) / Float64(sqrt(fma(Float64(1.0 - cos(Float64(2.0 * kx))), 2.0, Float64(Float64(fma(fma(Float64(ky * ky), 0.17777777777777778, -1.3333333333333333), Float64(ky * ky), 4.0) * ky) * ky))) / 2.0)) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 2.0)) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.4e-191], N[Power[N[(N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[ky, 0.68], N[(N[(N[(N[(N[(0.008333333333333333 * N[(ky * ky), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.17777777777777778 + -1.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 4.0), $MachinePrecision] * ky), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 2.3999999999999999e-191

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 28.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Applied rewrites 28.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   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. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sin kx} \]

      6. lower-*.f64 28.8

         \[\leadsto \frac{\color{blue}{\sin th \cdot \sin ky}}{\sin kx} \]

   7. Applied rewrites 28.8%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sin kx}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\sin kx}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin kx}{\sin th \cdot \sin ky}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sin kx}{\sin th \cdot \sin ky}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\sin kx}{\color{blue}{\sin th \cdot \sin ky}}} \]

      5. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}} \]

      7. lower-/.f64 28.4

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\sin kx}{\sin th}}}{\sin ky}} \]

   9. Applied rewrites 28.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sin kx}{\sin th}}{\sin ky}}} \]

   if 2.3999999999999999e-191 < ky < 0.680000000000000049

   1. Initial program 97.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 61.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 86.0%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) \cdot {ky}^{2}} + 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, {ky}^{2}, 1\right)} \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {ky}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} + \color{blue}{\frac{-1}{6}}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {ky}^{2}, \frac{-1}{6}\right)}, {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{ky \cdot ky}, \frac{-1}{6}\right), {ky}^{2}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, ky \cdot ky, \frac{-1}{6}\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{8}{45}, \frac{-4}{3}\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

      12. lower-*.f64 86.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), \color{blue}{ky \cdot ky}, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   10. Applied rewrites 86.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th \]

   if 0.680000000000000049 < ky

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right)} \cdot 2}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(1 - \color{blue}{\cos \left(2 \cdot ky\right)}\right) \cdot 2}}{2}} \cdot \sin th \]

      5. lower-*.f64 53.4

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \color{blue}{\left(2 \cdot ky\right)}\right) \cdot 2}}{2}} \cdot \sin th \]

   7. Applied rewrites 53.4%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}}{2}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 2.4 \cdot 10^{-191}:\\ \;\;\;\;{\left(\frac{\frac{\sin kx}{\sin th}}{\sin ky}\right)}^{-1}\\ \mathbf{elif}\;ky \leq 0.68:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, ky \cdot ky, -0.16666666666666666\right), ky \cdot ky, 1\right) \cdot ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}{2}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 37.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.15 \cdot 10^{-114}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.15e-114)
   (sin th)
   (if (<= kx 2.9e-6)
     (* (/ (sin ky) (sqrt (+ (* kx kx) (* ky ky)))) (sin th))
     (*
      (/ (sin ky) (/ (sqrt (* (- 1.0 (cos (* 2.0 kx))) 2.0)) 2.0))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.15e-114) {
		tmp = sin(th);
	} else if (kx <= 2.9e-6) {
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	} else {
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * kx))) * 2.0)) / 2.0)) * 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 (kx <= 1.15d-114) then
        tmp = sin(th)
    else if (kx <= 2.9d-6) then
        tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th)
    else
        tmp = (sin(ky) / (sqrt(((1.0d0 - cos((2.0d0 * kx))) * 2.0d0)) / 2.0d0)) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.15e-114) {
		tmp = Math.sin(th);
	} else if (kx <= 2.9e-6) {
		tmp = (Math.sin(ky) / Math.sqrt(((kx * kx) + (ky * ky)))) * Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / (Math.sqrt(((1.0 - Math.cos((2.0 * kx))) * 2.0)) / 2.0)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.15e-114:
		tmp = math.sin(th)
	elif kx <= 2.9e-6:
		tmp = (math.sin(ky) / math.sqrt(((kx * kx) + (ky * ky)))) * math.sin(th)
	else:
		tmp = (math.sin(ky) / (math.sqrt(((1.0 - math.cos((2.0 * kx))) * 2.0)) / 2.0)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.15e-114)
		tmp = sin(th);
	elseif (kx <= 2.9e-6)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + Float64(ky * ky)))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * 2.0)) / 2.0)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.15e-114)
		tmp = sin(th);
	elseif (kx <= 2.9e-6)
		tmp = (sin(ky) / sqrt(((kx * kx) + (ky * ky)))) * sin(th);
	else
		tmp = (sin(ky) / (sqrt(((1.0 - cos((2.0 * kx))) * 2.0)) / 2.0)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.15e-114], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 2.9e-6], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 1.15e-114

   1. Initial program 94.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 29.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Applied rewrites 29.6%

      \[\leadsto \color{blue}{\sin th} \]

   if 1.15e-114 < kx < 2.9000000000000002e-6

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.8

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Applied rewrites 99.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

      2. lower-*.f64 66.9

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   8. Applied rewrites 66.9%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   if 2.9000000000000002e-6 < kx

   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-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky} \cdot \sin ky + {\sin kx}^{2}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      8. sin-mult N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\sin kx} \cdot \sin kx}} \cdot \sin th \]

      12. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      13. sin-mult N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2} + \color{blue}{\frac{\cos \left(kx - kx\right) - \cos \left(kx + kx\right)}{2}}}} \cdot \sin th \]

      14. frac-add N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{2 \cdot 2}}}} \cdot \sin th \]

      15. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{4}}}} \cdot \sin th \]

      16. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}{\color{blue}{2 + 2}}}} \cdot \sin th \]

      17. sqrt-div N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(kx - kx\right) - \cos \left(kx + kx\right)\right)}}{\sqrt{2 + 2}}}} \cdot \sin th \]

   4. Applied rewrites 97.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, 2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)\right)}}{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}}{2}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2} + {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)}}{2}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}}{2}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(\color{blue}{1 - \cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \color{blue}{\left(2 \cdot kx\right)}, 2, {ky}^{2} \cdot \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right)\right)}}{2}} \cdot \sin th \]

      6. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot {ky}^{2}}\right)}}{2}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot \color{blue}{\left(ky \cdot ky\right)}\right)}}{2}} \cdot \sin th \]

      8. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \color{blue}{\left(\left(4 + {ky}^{2} \cdot \left(\frac{8}{45} \cdot {ky}^{2} - \frac{4}{3}\right)\right) \cdot ky\right) \cdot ky}\right)}}{2}} \cdot \sin th \]

   7. Applied rewrites 44.0%

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, \left(\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.17777777777777778, -1.3333333333333333\right), ky \cdot ky, 4\right) \cdot ky\right) \cdot ky\right)}}}{2}} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right)}}}{2}} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 52.5%

       \[\leadsto \frac{\sin ky}{\frac{\sqrt{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot \color{blue}{2}}}{2}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 13.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\mathsf{fma}\left(0.16666666666666666, th \cdot th, 1\right)}{th}\right)}^{-1} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (pow (/ (fma 0.16666666666666666 (* th th) 1.0) th) -1.0))

C

double code(double kx, double ky, double th) {
	return pow((fma(0.16666666666666666, (th * th), 1.0) / th), -1.0);
}

Julia

function code(kx, ky, th)
	return Float64(fma(0.16666666666666666, Float64(th * th), 1.0) / th) ^ -1.0
end

Wolfram

code[kx_, ky_, th_] := N[Power[N[(N[(0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] / th), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 96.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 23.6

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 23.6%

   \[\leadsto \color{blue}{\sin th} \]
6. Step-by-step derivation

7. Applied rewrites 11.7%

   \[\leadsto e^{\log \left({\sin th}^{-1}\right) \cdot -1} \]

8. Taylor expanded in th around 0

   \[\leadsto e^{\log \left(\frac{1 + \frac{1}{6} \cdot {th}^{2}}{th}\right) \cdot -1} \]
9. Step-by-step derivation

10. Applied rewrites 6.1%

    \[\leadsto e^{\log \left(\frac{\mathsf{fma}\left(th \cdot th, 0.16666666666666666, 1\right)}{th}\right) \cdot -1} \]
11. Step-by-step derivation

12. Applied rewrites 12.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, th \cdot th, 1\right)}{th}}} \]

13. Final simplification 12.6%

    \[\leadsto {\left(\frac{\mathsf{fma}\left(0.16666666666666666, th \cdot th, 1\right)}{th}\right)}^{-1} \]
14. Add Preprocessing

Alternative 30: 22.7% 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 96.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 23.6

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 23.6%

   \[\leadsto \color{blue}{\sin th} \]
6. Add Preprocessing

Alternative 31: 12.4% 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 96.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 23.6

      \[\leadsto \color{blue}{\sin th} \]

5. Applied rewrites 23.6%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 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. Taylor expanded in th around 0

    \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
12. Step-by-step derivation

13. Applied rewrites 11.9%

    \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024307 
(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)))