Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.4% → 99.7%

Time: 11.1s

Alternatives: 28

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

Alternative 2: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := \frac{t\_1}{\sin ky}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin kx}^{2}}}\\ t_5 := \frac{\sin th}{\frac{t\_1}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}\\ \mathbf{if}\;t\_4 \leq -0.99999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.42:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 0.02:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{t\_3}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky)))
        (t_2 (/ t_1 (sin ky)))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin kx) 2.0)))))
        (t_5
         (/ (sin th) (/ t_1 (* (fma -0.16666666666666666 (* ky ky) 1.0) ky)))))
   (if (<= t_4 -0.99999)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.42)
       (/ (* (fma (* th th) -0.16666666666666666 1.0) th) t_2)
       (if (<= t_4 0.02)
         t_5
         (if (<= t_4 0.995)
           (/
            (*
             (fma
              (fma (* th th) 0.008333333333333333 -0.16666666666666666)
              (* th th)
              1.0)
             th)
            t_2)
           (if (<= t_4 1.0)
             (fma (* -0.5 (* kx kx)) (/ (sin th) t_3) (sin th))
             t_5)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double t_2 = t_1 / sin(ky);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(kx), 2.0)));
	double t_5 = sin(th) / (t_1 / (fma(-0.16666666666666666, (ky * ky), 1.0) * ky));
	double tmp;
	if (t_4 <= -0.99999) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.42) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / t_2;
	} else if (t_4 <= 0.02) {
		tmp = t_5;
	} else if (t_4 <= 0.995) {
		tmp = (fma(fma((th * th), 0.008333333333333333, -0.16666666666666666), (th * th), 1.0) * th) / t_2;
	} else if (t_4 <= 1.0) {
		tmp = fma((-0.5 * (kx * kx)), (sin(th) / t_3), sin(th));
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	t_2 = Float64(t_1 / sin(ky))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(kx) ^ 2.0))))
	t_5 = Float64(sin(th) / Float64(t_1 / Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky)))
	tmp = 0.0
	if (t_4 <= -0.99999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.42)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / t_2);
	elseif (t_4 <= 0.02)
		tmp = t_5;
	elseif (t_4 <= 0.995)
		tmp = Float64(Float64(fma(fma(Float64(th * th), 0.008333333333333333, -0.16666666666666666), Float64(th * th), 1.0) * th) / t_2);
	elseif (t_4 <= 1.0)
		tmp = fma(Float64(-0.5 * Float64(kx * kx)), Float64(sin(th) / t_3), sin(th));
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.99999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.42], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.02], t$95$5, If[LessEqual[t$95$4, 0.995], N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 1.0], N[(N[(-0.5 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$3), $MachinePrecision] + N[Sin[th], $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]]]]

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))))) < -0.999990000000000046

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

   if -0.999990000000000046 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.419999999999999984

   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-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.3

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 59.5

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

   9. Applied rewrites 59.5%

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

   if -0.419999999999999984 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004 or 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 92.9

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

   9. Applied rewrites 92.9%

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

   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))))) < 0.994999999999999996

   1. Initial program 99.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) \cdot th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) \cdot th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right)} \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) \cdot {th}^{2}} + 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}, {th}^{2}, 1\right)} \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {th}^{2}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{th}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {th}^{2}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({th}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {th}^{2}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {th}^{2}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{120}, \frac{-1}{6}\right), {th}^{2}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{1}{120}, \frac{-1}{6}\right), {th}^{2}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{th \cdot th}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      13. lower-*.f64 55.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), \color{blue}{th \cdot th}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

   9. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 100.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 95.5

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

   7. Applied rewrites 95.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.99999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.42:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{{\sin ky}^{2}}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := \frac{\sin th}{\frac{t\_1}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin kx}^{2}}}\\ t_5 := \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{t\_1}{\sin ky}}\\ \mathbf{if}\;t\_4 \leq -0.99999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.42:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{t\_3}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky)))
        (t_2
         (/ (sin th) (/ t_1 (* (fma -0.16666666666666666 (* ky ky) 1.0) ky))))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin kx) 2.0)))))
        (t_5
         (/ (* (fma (* th th) -0.16666666666666666 1.0) th) (/ t_1 (sin ky)))))
   (if (<= t_4 -0.99999)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.42)
       t_5
       (if (<= t_4 0.02)
         t_2
         (if (<= t_4 0.995)
           t_5
           (if (<= t_4 1.0)
             (fma (* -0.5 (* kx kx)) (/ (sin th) t_3) (sin th))
             t_2)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double t_2 = sin(th) / (t_1 / (fma(-0.16666666666666666, (ky * ky), 1.0) * ky));
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(kx), 2.0)));
	double t_5 = (fma((th * th), -0.16666666666666666, 1.0) * th) / (t_1 / sin(ky));
	double tmp;
	if (t_4 <= -0.99999) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.42) {
		tmp = t_5;
	} else if (t_4 <= 0.02) {
		tmp = t_2;
	} else if (t_4 <= 0.995) {
		tmp = t_5;
	} else if (t_4 <= 1.0) {
		tmp = fma((-0.5 * (kx * kx)), (sin(th) / t_3), sin(th));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	t_2 = Float64(sin(th) / Float64(t_1 / Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky)))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(kx) ^ 2.0))))
	t_5 = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(t_1 / sin(ky)))
	tmp = 0.0
	if (t_4 <= -0.99999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.42)
		tmp = t_5;
	elseif (t_4 <= 0.02)
		tmp = t_2;
	elseif (t_4 <= 0.995)
		tmp = t_5;
	elseif (t_4 <= 1.0)
		tmp = fma(Float64(-0.5 * Float64(kx * kx)), Float64(sin(th) / t_3), sin(th));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.99999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.42], t$95$5, If[LessEqual[t$95$4, 0.02], t$95$2, If[LessEqual[t$95$4, 0.995], t$95$5, If[LessEqual[t$95$4, 1.0], N[(N[(-0.5 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$3), $MachinePrecision] + N[Sin[th], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]]

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))))) < -0.999990000000000046

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

   if -0.999990000000000046 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.419999999999999984 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))))) < 0.994999999999999996

   1. Initial program 99.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 56.8

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

   9. Applied rewrites 56.8%

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

   if -0.419999999999999984 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004 or 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 92.9

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

   9. Applied rewrites 92.9%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 100.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 95.5

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

   7. Applied rewrites 95.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.99999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.42:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{{\sin ky}^{2}}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := \left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky\right) \cdot \frac{\sin th}{t\_1}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin kx}^{2}}}\\ t_5 := \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{t\_1}{\sin ky}}\\ \mathbf{if}\;t\_4 \leq -0.99999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.42:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{t\_3}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky)))
        (t_2
         (* (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (/ (sin th) t_1)))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin kx) 2.0)))))
        (t_5
         (/ (* (fma (* th th) -0.16666666666666666 1.0) th) (/ t_1 (sin ky)))))
   (if (<= t_4 -0.99999)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.42)
       t_5
       (if (<= t_4 0.02)
         t_2
         (if (<= t_4 0.995)
           t_5
           (if (<= t_4 1.0)
             (fma (* -0.5 (* kx kx)) (/ (sin th) t_3) (sin th))
             t_2)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double t_2 = (fma(-0.16666666666666666, (ky * ky), 1.0) * ky) * (sin(th) / t_1);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(kx), 2.0)));
	double t_5 = (fma((th * th), -0.16666666666666666, 1.0) * th) / (t_1 / sin(ky));
	double tmp;
	if (t_4 <= -0.99999) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.42) {
		tmp = t_5;
	} else if (t_4 <= 0.02) {
		tmp = t_2;
	} else if (t_4 <= 0.995) {
		tmp = t_5;
	} else if (t_4 <= 1.0) {
		tmp = fma((-0.5 * (kx * kx)), (sin(th) / t_3), sin(th));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	t_2 = Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) * Float64(sin(th) / t_1))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(kx) ^ 2.0))))
	t_5 = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(t_1 / sin(ky)))
	tmp = 0.0
	if (t_4 <= -0.99999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.42)
		tmp = t_5;
	elseif (t_4 <= 0.02)
		tmp = t_2;
	elseif (t_4 <= 0.995)
		tmp = t_5;
	elseif (t_4 <= 1.0)
		tmp = fma(Float64(-0.5 * Float64(kx * kx)), Float64(sin(th) / t_3), sin(th));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.99999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.42], t$95$5, If[LessEqual[t$95$4, 0.02], t$95$2, If[LessEqual[t$95$4, 0.995], t$95$5, If[LessEqual[t$95$4, 1.0], N[(N[(-0.5 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$3), $MachinePrecision] + N[Sin[th], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]]

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))))) < -0.999990000000000046

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

   if -0.999990000000000046 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.419999999999999984 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))))) < 0.994999999999999996

   1. Initial program 99.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 56.8

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

   9. Applied rewrites 56.8%

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

   if -0.419999999999999984 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004 or 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lower-/.f64 99.6

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

      11. lift-sin.f64 N/A

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

      12. lift-hypot.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-hypot.f64 N/A

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

      15. lift-sin.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in ky around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 92.9

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

   11. Applied rewrites 92.9%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 100.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 95.5

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

   7. Applied rewrites 95.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.99999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.42:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{{\sin ky}^{2}}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := \left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky\right) \cdot \frac{\sin th}{t\_1}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin kx}^{2}}}\\ t_5 := \frac{\sin ky}{t\_1} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{if}\;t\_4 \leq -0.99999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.42:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{t\_3}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky)))
        (t_2
         (* (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (/ (sin th) t_1)))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin kx) 2.0)))))
        (t_5
         (* (/ (sin ky) t_1) (* (fma (* th th) -0.16666666666666666 1.0) th))))
   (if (<= t_4 -0.99999)
     (* (/ (sin ky) (sqrt (+ (* kx kx) t_3))) (sin th))
     (if (<= t_4 -0.42)
       t_5
       (if (<= t_4 0.02)
         t_2
         (if (<= t_4 0.995)
           t_5
           (if (<= t_4 1.0)
             (fma (* -0.5 (* kx kx)) (/ (sin th) t_3) (sin th))
             t_2)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double t_2 = (fma(-0.16666666666666666, (ky * ky), 1.0) * ky) * (sin(th) / t_1);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(kx), 2.0)));
	double t_5 = (sin(ky) / t_1) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	double tmp;
	if (t_4 <= -0.99999) {
		tmp = (sin(ky) / sqrt(((kx * kx) + t_3))) * sin(th);
	} else if (t_4 <= -0.42) {
		tmp = t_5;
	} else if (t_4 <= 0.02) {
		tmp = t_2;
	} else if (t_4 <= 0.995) {
		tmp = t_5;
	} else if (t_4 <= 1.0) {
		tmp = fma((-0.5 * (kx * kx)), (sin(th) / t_3), sin(th));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	t_2 = Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) * Float64(sin(th) / t_1))
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(kx) ^ 2.0))))
	t_5 = Float64(Float64(sin(ky) / t_1) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th))
	tmp = 0.0
	if (t_4 <= -0.99999)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(kx * kx) + t_3))) * sin(th));
	elseif (t_4 <= -0.42)
		tmp = t_5;
	elseif (t_4 <= 0.02)
		tmp = t_2;
	elseif (t_4 <= 0.995)
		tmp = t_5;
	elseif (t_4 <= 1.0)
		tmp = fma(Float64(-0.5 * Float64(kx * kx)), Float64(sin(th) / t_3), sin(th));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.99999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(kx * kx), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.42], t$95$5, If[LessEqual[t$95$4, 0.02], t$95$2, If[LessEqual[t$95$4, 0.995], t$95$5, If[LessEqual[t$95$4, 1.0], N[(N[(-0.5 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$3), $MachinePrecision] + N[Sin[th], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]]

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))))) < -0.999990000000000046

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

   if -0.999990000000000046 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.419999999999999984 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))))) < 0.994999999999999996

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. cos-+PI/2-rev N/A

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

      6. lift-sin.f64 N/A

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

      7. cos-+PI/2-rev N/A

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

      8. 1-sub-sin-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sin-+PI/2-rev N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

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

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 20.9

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 20.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 12.9

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

   10. Applied rewrites 12.9%

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

   11. Taylor expanded in kx around inf

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. unpow2 N/A

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

       4. 1-sub-cos N/A

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

       5. unpow2 N/A

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

       6. unpow2 N/A

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

       7. unpow2 N/A

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

       8. lower-hypot.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-sin.f64 56.6

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

   13. Applied rewrites 56.6%

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

   if -0.419999999999999984 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004 or 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lower-/.f64 99.6

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

      11. lift-sin.f64 N/A

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

      12. lift-hypot.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-hypot.f64 N/A

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

      15. lift-sin.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in ky around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 92.9

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

   11. Applied rewrites 92.9%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 100.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 95.5

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

   7. Applied rewrites 95.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.99999:\\ \;\;\;\;\frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.42:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{{\sin ky}^{2}}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin kx}^{2}}}\\ t_3 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_4 := \left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky\right) \cdot \frac{\sin th}{t\_3}\\ t_5 := \frac{\sin ky}{t\_3} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{if}\;t\_2 \leq -0.99999:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.42:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{t\_1}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin kx) 2.0)))))
        (t_3 (hypot (sin kx) (sin ky)))
        (t_4
         (* (* (fma -0.16666666666666666 (* ky ky) 1.0) ky) (/ (sin th) t_3)))
        (t_5
         (* (/ (sin ky) t_3) (* (fma (* th th) -0.16666666666666666 1.0) th))))
   (if (<= t_2 -0.99999)
     (* (/ (sin ky) (fabs (sin ky))) (sin th))
     (if (<= t_2 -0.42)
       t_5
       (if (<= t_2 0.02)
         t_4
         (if (<= t_2 0.995)
           t_5
           (if (<= t_2 1.0)
             (fma (* -0.5 (* kx kx)) (/ (sin th) t_1) (sin th))
             t_4)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(kx), 2.0)));
	double t_3 = hypot(sin(kx), sin(ky));
	double t_4 = (fma(-0.16666666666666666, (ky * ky), 1.0) * ky) * (sin(th) / t_3);
	double t_5 = (sin(ky) / t_3) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	double tmp;
	if (t_2 <= -0.99999) {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	} else if (t_2 <= -0.42) {
		tmp = t_5;
	} else if (t_2 <= 0.02) {
		tmp = t_4;
	} else if (t_2 <= 0.995) {
		tmp = t_5;
	} else if (t_2 <= 1.0) {
		tmp = fma((-0.5 * (kx * kx)), (sin(th) / t_1), sin(th));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(kx) ^ 2.0))))
	t_3 = hypot(sin(kx), sin(ky))
	t_4 = Float64(Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky) * Float64(sin(th) / t_3))
	t_5 = Float64(Float64(sin(ky) / t_3) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th))
	tmp = 0.0
	if (t_2 <= -0.99999)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	elseif (t_2 <= -0.42)
		tmp = t_5;
	elseif (t_2 <= 0.02)
		tmp = t_4;
	elseif (t_2 <= 0.995)
		tmp = t_5;
	elseif (t_2 <= 1.0)
		tmp = fma(Float64(-0.5 * Float64(kx * kx)), Float64(sin(th) / t_1), sin(th));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Sin[ky], $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.99999], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.42], t$95$5, If[LessEqual[t$95$2, 0.02], t$95$4, If[LessEqual[t$95$2, 0.995], t$95$5, If[LessEqual[t$95$2, 1.0], N[(N[(-0.5 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / t$95$1), $MachinePrecision] + N[Sin[th], $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.999990000000000046

   1. Initial program 91.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. cos-+PI/2-rev N/A

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

      6. lift-sin.f64 N/A

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

      7. cos-+PI/2-rev N/A

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

      8. 1-sub-sin-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sin-+PI/2-rev N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

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

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 93.8

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 93.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   if -0.999990000000000046 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.419999999999999984 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))))) < 0.994999999999999996

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. cos-+PI/2-rev N/A

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

      6. lift-sin.f64 N/A

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

      7. cos-+PI/2-rev N/A

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

      8. 1-sub-sin-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sin-+PI/2-rev N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

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

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 20.9

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 20.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 12.9

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

   10. Applied rewrites 12.9%

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

   11. Taylor expanded in kx around inf

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. unpow2 N/A

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

       4. 1-sub-cos N/A

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

       5. unpow2 N/A

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

       6. unpow2 N/A

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

       7. unpow2 N/A

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

       8. lower-hypot.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-sin.f64 56.6

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

   13. Applied rewrites 56.6%

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

   if -0.419999999999999984 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004 or 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 89.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-hypot.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lower-/.f64 99.6

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

      11. lift-sin.f64 N/A

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

      12. lift-hypot.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-hypot.f64 N/A

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

      15. lift-sin.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in ky around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 92.9

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

   11. Applied rewrites 92.9%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1

   1. Initial program 100.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 95.5

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

   7. Applied rewrites 95.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.99999:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.42:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{{\sin ky}^{2}}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.62:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.08:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.62)
     (* (/ (sin ky) (fabs (sin ky))) (sin th))
     (if (<= t_1 0.08) (/ (sin th) (/ (sin kx) (sin ky))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.62) {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	} else if (t_1 <= 0.08) {
		tmp = sin(th) / (sin(kx) / sin(ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.62d0)) then
        tmp = (sin(ky) / abs(sin(ky))) * sin(th)
    else if (t_1 <= 0.08d0) then
        tmp = sin(th) / (sin(kx) / sin(ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.62) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(ky))) * Math.sin(th);
	} else if (t_1 <= 0.08) {
		tmp = Math.sin(th) / (Math.sin(kx) / Math.sin(ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.62:
		tmp = (math.sin(ky) / math.fabs(math.sin(ky))) * math.sin(th)
	elif t_1 <= 0.08:
		tmp = math.sin(th) / (math.sin(kx) / math.sin(ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.62)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	elseif (t_1 <= 0.08)
		tmp = Float64(sin(th) / Float64(sin(kx) / sin(ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.62)
		tmp = (sin(ky) / abs(sin(ky))) * sin(th);
	elseif (t_1 <= 0.08)
		tmp = sin(th) / (sin(kx) / sin(ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.62], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.08], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.619999999999999996

   1. Initial program 93.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. cos-+PI/2-rev N/A

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

      6. lift-sin.f64 N/A

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

      7. cos-+PI/2-rev N/A

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

      8. 1-sub-sin-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sin-+PI/2-rev N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 79.0

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

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

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 75.8

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 75.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   if -0.619999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0800000000000000017

   1. Initial program 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in ky around 0

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

      1. lower-sin.f64 63.3

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

   9. Applied rewrites 63.3%

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

   if 0.0800000000000000017 < (/.f64 (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 90.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.9

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

   7. Applied rewrites 59.9%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.62:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.08:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.62:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.08:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.62)
     (* (/ (sin ky) (fabs (sin ky))) (sin th))
     (if (<= t_1 0.08) (* (/ (sin ky) (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.62) {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	} else if (t_1 <= 0.08) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.62d0)) then
        tmp = (sin(ky) / abs(sin(ky))) * sin(th)
    else if (t_1 <= 0.08d0) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.62) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(ky))) * Math.sin(th);
	} else if (t_1 <= 0.08) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.62:
		tmp = (math.sin(ky) / math.fabs(math.sin(ky))) * math.sin(th)
	elif t_1 <= 0.08:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.62)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	elseif (t_1 <= 0.08)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.62)
		tmp = (sin(ky) / abs(sin(ky))) * sin(th);
	elseif (t_1 <= 0.08)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.62], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.08], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.619999999999999996

   1. Initial program 93.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. cos-+PI/2-rev N/A

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

      6. lift-sin.f64 N/A

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

      7. cos-+PI/2-rev N/A

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

      8. 1-sub-sin-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sin-+PI/2-rev N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 79.0

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

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

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 75.8

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 75.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   if -0.619999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0800000000000000017

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 63.4

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

   5. Applied rewrites 63.4%

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

   if 0.0800000000000000017 < (/.f64 (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 90.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.9

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

   7. Applied rewrites 59.9%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.62:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.08:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.62:\\ \;\;\;\;\frac{\sin th}{\left|\sin ky\right|} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.08:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.62)
     (* (/ (sin th) (fabs (sin ky))) (sin ky))
     (if (<= t_1 0.08) (* (/ (sin ky) (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.62) {
		tmp = (sin(th) / fabs(sin(ky))) * sin(ky);
	} else if (t_1 <= 0.08) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))
    if (t_1 <= (-0.62d0)) then
        tmp = (sin(th) / abs(sin(ky))) * sin(ky)
    else if (t_1 <= 0.08d0) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.62) {
		tmp = (Math.sin(th) / Math.abs(Math.sin(ky))) * Math.sin(ky);
	} else if (t_1 <= 0.08) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))
	tmp = 0
	if t_1 <= -0.62:
		tmp = (math.sin(th) / math.fabs(math.sin(ky))) * math.sin(ky)
	elif t_1 <= 0.08:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.62)
		tmp = Float64(Float64(sin(th) / abs(sin(ky))) * sin(ky));
	elseif (t_1 <= 0.08)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.62)
		tmp = (sin(th) / abs(sin(ky))) * sin(ky);
	elseif (t_1 <= 0.08)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.62], N[(N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.08], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.619999999999999996

   1. Initial program 93.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. cos-+PI/2-rev N/A

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

      6. lift-sin.f64 N/A

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

      7. cos-+PI/2-rev N/A

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

      8. 1-sub-sin-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sin-+PI/2-rev N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 79.0

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

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

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 75.8

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 75.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\left|\sin ky\right|}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\left|\sin ky\right|}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}} \]

      6. lower-/.f64 75.6

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\left|\sin ky\right|}} \]

   9. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\left|\sin ky\right|}} \]

   if -0.619999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0800000000000000017

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 63.4

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

   5. Applied rewrites 63.4%

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

   if 0.0800000000000000017 < (/.f64 (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 90.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.9

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

   7. Applied rewrites 59.9%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.62:\\ \;\;\;\;\frac{\sin th}{\left|\sin ky\right|} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.08:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.75:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot th}{\left|\sin ky\right|} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.08:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.75)
     (*
      (/ (* (fma -0.16666666666666666 (* th th) 1.0) th) (fabs (sin ky)))
      (sin ky))
     (if (<= t_1 0.08) (* (/ (sin ky) (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.75) {
		tmp = ((fma(-0.16666666666666666, (th * th), 1.0) * th) / fabs(sin(ky))) * sin(ky);
	} else if (t_1 <= 0.08) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.75)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(th * th), 1.0) * th) / abs(sin(ky))) * sin(ky));
	elseif (t_1 <= 0.08)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * 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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.75], N[(N[(N[(N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.08], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.75

   1. Initial program 93.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. cos-+PI/2-rev N/A

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

      6. lift-sin.f64 N/A

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

      7. cos-+PI/2-rev N/A

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

      8. 1-sub-sin-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sin-+PI/2-rev N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

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

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 81.1

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 81.1%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 39.0

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

   10. Applied rewrites 39.0%

       \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 39.0

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

   12. Applied rewrites 39.0%

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

   if -0.75 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0800000000000000017

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

   if 0.0800000000000000017 < (/.f64 (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 90.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.9

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

   7. Applied rewrites 59.9%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.75:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot th}{\left|\sin ky\right|} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.08:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot th}{\left|\sin ky\right|} \cdot \sin ky\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))))
   (if (<= t_1 -0.1)
     (*
      (/ (* (fma -0.16666666666666666 (* th th) 1.0) th) (fabs (sin ky)))
      (sin ky))
     (if (<= t_1 0.02) (/ (sin th) (/ (sin kx) ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = ((fma(-0.16666666666666666, (th * th), 1.0) * th) / fabs(sin(ky))) * sin(ky);
	} else if (t_1 <= 0.02) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(th * th), 1.0) * th) / abs(sin(ky))) * sin(ky));
	elseif (t_1 <= 0.02)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[(N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. cos-+PI/2-rev N/A

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

      6. lift-sin.f64 N/A

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

      7. cos-+PI/2-rev N/A

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

      8. 1-sub-sin-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sin-+PI/2-rev N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 81.2

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

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

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 69.4

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 69.4%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 33.6

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

   10. Applied rewrites 33.6%

       \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 33.6

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

   12. Applied rewrites 33.6%

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

   if -0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0200000000000000004

   1. Initial program 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 71.7

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

   9. Applied rewrites 71.7%

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

   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 90.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.0

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

   7. Applied rewrites 59.0%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq -0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot th}{\left|\sin ky\right|} \cdot \sin ky\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.02)
   (/ (sin th) (/ (sin kx) ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.02) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.02d0) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.02) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.02:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.02)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.02)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 33.1

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

   9. Applied rewrites 33.1%

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

   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 90.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.0

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

   7. Applied rewrites 59.0%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 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(ky), 2.0) + pow(sin(kx), 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(ky) ** 2.0d0) + (sin(kx) ** 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(ky), 2.0) + Math.pow(Math.sin(kx), 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(ky), 2.0) + math.pow(math.sin(kx), 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(ky) ^ 2.0) + (sin(kx) ^ 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(ky) ^ 2.0) + (sin(kx) ^ 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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $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 96.5%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 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 90.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.0

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

   7. Applied rewrites 59.0%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 0.02)
   (/ (* ky (sin th)) (sin kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 0.02) {
		tmp = (ky * sin(th)) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 0.02d0) then
        tmp = (ky * sin(th)) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 0.02) {
		tmp = (ky * Math.sin(th)) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 0.02:
		tmp = (ky * math.sin(th)) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.02)
		tmp = Float64(Float64(ky * sin(th)) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 0.02)
		tmp = (ky * sin(th)) / sin(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(ky * N[Sin[th], $MachinePrecision]), $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 96.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   8. 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 31.5

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

   9. Applied rewrites 31.5%

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

   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 90.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sin.f64 59.0

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

   7. Applied rewrites 59.0%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))) 2e-17)
   (* (/ ky (sin kx)) (* (fma (* th th) -0.16666666666666666 1.0) th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 2e-17) {
		tmp = (ky / sin(kx)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 2e-17)
		tmp = Float64(Float64(ky / sin(kx)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * 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[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-17], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * 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))))) < 2.00000000000000014e-17

   1. Initial program 96.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. sqr-neg-rev N/A

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

      4. lift-sin.f64 N/A

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

      5. cos-+PI/2-rev N/A

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

      6. lift-sin.f64 N/A

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

      7. cos-+PI/2-rev N/A

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

      8. 1-sub-sin-rev N/A

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

      9. sin-+PI/2-rev N/A

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

      10. sin-+PI/2-rev N/A

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

      11. lower--.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in kx around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

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

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 42.0

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 42.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 21.0

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

   10. Applied rewrites 21.0%

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

   11. Taylor expanded in ky around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sin.f64 21.5

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

   13. Applied rewrites 21.5%

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

   if 2.00000000000000014e-17 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-sin.f64 58.0

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

   7. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 15.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sin th \leq 10^{-317}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))
       (sin th))
      1e-317)
   (* (* (* th th) -0.16666666666666666) th)
   (*
    (fma
     (fma (* th th) 0.008333333333333333 -0.16666666666666666)
     (* th th)
     1.0)
    th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) * sin(th)) <= 1e-317) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = fma(fma((th * th), 0.008333333333333333, -0.16666666666666666), (th * th), 1.0) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) * sin(th)) <= 1e-317)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * th);
	else
		tmp = Float64(fma(fma(Float64(th * th), 0.008333333333333333, -0.16666666666666666), Float64(th * th), 1.0) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 1e-317], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 1.00000023e-317

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. lower-sin.f64 19.7

         \[\leadsto \color{blue}{\sin th} \]

   7. Applied rewrites 19.7%

      \[\leadsto \color{blue}{\sin th} \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 10.1%

       \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   11. Taylor expanded in th around inf

       \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 15.5%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   if 1.00000023e-317 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 96.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. lower-sin.f64 29.9

         \[\leadsto \color{blue}{\sin th} \]

   7. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\sin th} \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 15.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot \color{blue}{th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sin th \leq 10^{-317}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(th \cdot th, 0.008333333333333333, -0.16666666666666666\right), th \cdot th, 1\right) \cdot th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 15.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sin th \leq 10^{-317}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))
       (sin th))
      1e-317)
   (* (* (* th th) -0.16666666666666666) th)
   (* (fma (* -0.16666666666666666 th) th 1.0) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) * sin(th)) <= 1e-317) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = fma((-0.16666666666666666 * th), th, 1.0) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) * sin(th)) <= 1e-317)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * th);
	else
		tmp = Float64(fma(Float64(-0.16666666666666666 * th), th, 1.0) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 1e-317], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th + 1.0), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 1.00000023e-317

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. lower-sin.f64 19.7

         \[\leadsto \color{blue}{\sin th} \]

   7. Applied rewrites 19.7%

      \[\leadsto \color{blue}{\sin th} \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 10.1%

       \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   11. Taylor expanded in th around inf

       \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 15.5%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   if 1.00000023e-317 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 96.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. lower-sin.f64 29.9

         \[\leadsto \color{blue}{\sin th} \]

   7. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\sin th} \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 15.9%

       \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.9%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \cdot th \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \sin th \leq 10^{-317}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot th, th, 1\right) \cdot th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 30.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2.95 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (/ (sin ky) (sqrt (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))))
      2.95e-30)
   (* (* (* th th) -0.16666666666666666) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))) <= 2.95e-30) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)))) <= 2.95d-30) then
        tmp = ((th * th) * (-0.16666666666666666d0)) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)))) <= 2.95e-30) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)))) <= 2.95e-30:
		tmp = ((th * th) * -0.16666666666666666) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 2.95e-30)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)))) <= 2.95e-30)
		tmp = ((th * th) * -0.16666666666666666) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.95e-30], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.9499999999999999e-30

   1. Initial program 96.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. lower-sin.f64 3.7

         \[\leadsto \color{blue}{\sin th} \]

   7. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\sin th} \]

   8. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 3.8%

       \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   11. Taylor expanded in th around inf

       \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   12. Step-by-step derivation

   13. Applied rewrites 14.4%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   if 2.9499999999999999e-30 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   6. Step-by-step derivation

      1. lower-sin.f64 57.5

         \[\leadsto \color{blue}{\sin th} \]

   7. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}}} \leq 2.95 \cdot 10^{-30}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (hypot (sin ky) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   3. +-commutative N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

   5. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

   7. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

   8. lower-hypot.f64 99.7

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

4. Applied rewrites 99.7%

   \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
5. Add Preprocessing

Alternative 20: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin th) (hypot (sin kx) (sin ky))) (sin ky)))

C

double code(double kx, double ky, double th) {
	return (sin(th) / hypot(sin(kx), sin(ky))) * sin(ky);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(th) / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
}

Python

def code(kx, ky, th):
	return (math.sin(th) / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(th) / hypot(sin(kx), sin(ky))) * sin(ky))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(th) / hypot(sin(kx), sin(ky))) * sin(ky);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   3. +-commutative N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

   5. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

   7. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

   8. lower-hypot.f64 99.7

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

4. Applied rewrites 99.7%

   \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

7. Final simplification 99.6%

   \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky \]
8. Add Preprocessing

Alternative 21: 75.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.0016:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(\left(1 - \cos \left(-2 \cdot ky\right)\right) + \left(1 - \cos \left(-2 \cdot kx\right)\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.0016)
   (/
    (sin th)
    (/
     (hypot (sin kx) (sin ky))
     (* (fma -0.16666666666666666 (* ky ky) 1.0) ky)))
   (*
    (/
     (sin ky)
     (/
      (sqrt (* (+ (- 1.0 (cos (* -2.0 ky))) (- 1.0 (cos (* -2.0 kx)))) 2.0))
      2.0))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.0016) {
		tmp = sin(th) / (hypot(sin(kx), sin(ky)) / (fma(-0.16666666666666666, (ky * ky), 1.0) * ky));
	} else {
		tmp = (sin(ky) / (sqrt((((1.0 - cos((-2.0 * ky))) + (1.0 - cos((-2.0 * kx)))) * 2.0)) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.0016)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), sin(ky)) / Float64(fma(-0.16666666666666666, Float64(ky * ky), 1.0) * ky)));
	else
		tmp = Float64(Float64(sin(ky) / Float64(sqrt(Float64(Float64(Float64(1.0 - cos(Float64(-2.0 * ky))) + Float64(1.0 - cos(Float64(-2.0 * kx)))) * 2.0)) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.0016], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[Sqrt[N[(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] * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.00160000000000000008

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot ky}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)} \cdot ky}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {ky}^{2}, 1\right)} \cdot ky}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}} \]

      6. lower-*.f64 61.5

         \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(-0.16666666666666666, \color{blue}{ky \cdot ky}, 1\right) \cdot ky}} \]

   9. Applied rewrites 61.5%

      \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}} \]

   if 0.00160000000000000008 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Applied rewrites 97.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, 2 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot 2 + 2 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)}}}{2}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)} + 2 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)}}{2}} \cdot \sin th \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right) + \color{blue}{2 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)}}}{2}} \cdot \sin th \]

      4. distribute-lft-out N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}}{2}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{2 \cdot \left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}}{2}} \cdot \sin th \]

      6. lower-+.f64 97.9

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \color{blue}{\left(\left(1 - \cos \left(2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}}{2}} \cdot \sin th \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \left(\left(1 - \color{blue}{\cos \left(2 \cdot kx\right)}\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}} \cdot \sin th \]

      8. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \left(\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}} \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \left(\left(1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \left(\left(1 - \cos \left(\mathsf{neg}\left(\color{blue}{2 \cdot kx}\right)\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}} \cdot \sin th \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \left(\left(1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}} \cdot \sin th \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \left(\left(1 - \cos \left(\color{blue}{-2} \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}} \cdot \sin th \]

      13. lower-*.f64 97.9

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{2 \cdot \left(\left(1 - \cos \color{blue}{\left(-2 \cdot kx\right)}\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}} \cdot \sin th \]

   8. Applied rewrites 97.9%

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{2 \cdot \left(\left(1 - \cos \left(-2 \cdot kx\right)\right) + \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}}{2}} \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 0.0016:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\mathsf{fma}\left(-0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{\left(\left(1 - \cos \left(-2 \cdot ky\right)\right) + \left(1 - \cos \left(-2 \cdot kx\right)\right)\right) \cdot 2}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 61.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.0038:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.0038)
   (*
    (/ (sin ky) (hypot (sin kx) (sin ky)))
    (* (fma (* th th) -0.16666666666666666 1.0) th))
   (* (/ (sin ky) (fabs (sin ky))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0038) {
		tmp = (sin(ky) / hypot(sin(kx), sin(ky))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.0038)
		tmp = Float64(Float64(sin(ky) / hypot(sin(kx), sin(ky))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	else
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.0038], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.00379999999999999999

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 87.3

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 87.3%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - {\cos ky}^{2}}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 44.8

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 44.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   8. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \cdot th\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \left(\left(\color{blue}{{th}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot th\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \left(\color{blue}{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right)} \cdot th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, \frac{-1}{6}, 1\right) \cdot th\right) \]

      7. lower-*.f64 29.6

         \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \left(\mathsf{fma}\left(\color{blue}{th \cdot th}, -0.16666666666666666, 1\right) \cdot th\right) \]

   10. Applied rewrites 29.6%

       \[\leadsto \frac{\sin ky}{\left|\sin ky\right|} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]

   11. Taylor expanded in kx around inf

       \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\left(1 + {\sin kx}^{2}\right) - {\cos ky}^{2}}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left({\sin kx}^{2} + 1\right)} - {\cos ky}^{2}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       2. associate--l+ N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + \left(1 - {\cos ky}^{2}\right)}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       3. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       4. 1-sub-cos N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       5. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       7. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       8. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin kx}, \sin ky\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       10. lower-sin.f64 65.7

           \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \color{blue}{\sin ky}\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   13. Applied rewrites 65.7%

       \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   if 0.00379999999999999999 < th

   1. Initial program 95.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 82.5

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 82.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - {\cos ky}^{2}}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 58.7

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 58.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 59.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;kx \leq 1.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0031746031746031746, kx \cdot kx, 0.044444444444444446\right), kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{2} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8.8e-159)
   (* (/ (sin ky) (fabs (sin ky))) (sin th))
   (if (<= kx 1.45)
     (*
      (/
       (sin ky)
       (sqrt
        (+
         (- 0.5 (* (cos (* 2.0 ky)) 0.5))
         (*
          (fma
           (fma
            (fma -0.0031746031746031746 (* kx kx) 0.044444444444444446)
            (* kx kx)
            -0.3333333333333333)
           (* kx kx)
           1.0)
          (* kx kx)))))
      (sin th))
     (*
      (/ (sin ky) (/ (* (sqrt 2.0) (sqrt (- 1.0 (cos (* -2.0 kx))))) 2.0))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8.8e-159) {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	} else if (kx <= 1.45) {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * ky)) * 0.5)) + (fma(fma(fma(-0.0031746031746031746, (kx * kx), 0.044444444444444446), (kx * kx), -0.3333333333333333), (kx * kx), 1.0) * (kx * kx))))) * sin(th);
	} else {
		tmp = (sin(ky) / ((sqrt(2.0) * sqrt((1.0 - cos((-2.0 * kx))))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8.8e-159)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	elseif (kx <= 1.45)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5)) + Float64(fma(fma(fma(-0.0031746031746031746, Float64(kx * kx), 0.044444444444444446), Float64(kx * kx), -0.3333333333333333), Float64(kx * kx), 1.0) * Float64(kx * kx))))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 - cos(Float64(-2.0 * kx))))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8.8e-159], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 1.45], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-0.0031746031746031746 * N[(kx * kx), $MachinePrecision] + 0.044444444444444446), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 8.8e-159

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 79.6

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 79.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - {\cos ky}^{2}}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 56.5

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 56.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   if 8.8e-159 < kx < 1.44999999999999996

   1. Initial program 98.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 97.6

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 97.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot {kx}^{2}} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot \color{blue}{\left(kx \cdot kx\right)} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right)} \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{\left(\frac{-1}{3} \cdot {kx}^{2} + 1\right)} \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {kx}^{2}, 1\right)} \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      9. lower-*.f64 95.1

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

   7. Applied rewrites 95.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      5. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      6. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      8. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky + ky\right)}\right)}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 95.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   9. Applied rewrites 95.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   10. Taylor expanded in kx around 0

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right)} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right) \cdot {kx}^{2}} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right)\right) \cdot {kx}^{2}} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       3. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left({kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right) + 1\right)} \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       4. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{\left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}\right) \cdot {kx}^{2}} + 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left({kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) - \frac{1}{3}, {kx}^{2}, 1\right)} \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       6. sub-neg N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{{kx}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       7. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) \cdot {kx}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       8. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}\right) \cdot {kx}^{2} + \color{blue}{\frac{-1}{3}}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{45} + \frac{-1}{315} \cdot {kx}^{2}, {kx}^{2}, \frac{-1}{3}\right)}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       10. +-commutative N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{315} \cdot {kx}^{2} + \frac{2}{45}}, {kx}^{2}, \frac{-1}{3}\right), {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       11. lower-fma.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{315}, {kx}^{2}, \frac{2}{45}\right)}, {kx}^{2}, \frac{-1}{3}\right), {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       12. unpow2 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{315}, \color{blue}{kx \cdot kx}, \frac{2}{45}\right), {kx}^{2}, \frac{-1}{3}\right), {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       13. lower-*.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{315}, \color{blue}{kx \cdot kx}, \frac{2}{45}\right), {kx}^{2}, \frac{-1}{3}\right), {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       14. unpow2 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{315}, kx \cdot kx, \frac{2}{45}\right), \color{blue}{kx \cdot kx}, \frac{-1}{3}\right), {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       15. lower-*.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{315}, kx \cdot kx, \frac{2}{45}\right), \color{blue}{kx \cdot kx}, \frac{-1}{3}\right), {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       16. unpow2 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{315}, kx \cdot kx, \frac{2}{45}\right), kx \cdot kx, \frac{-1}{3}\right), \color{blue}{kx \cdot kx}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{315}, kx \cdot kx, \frac{2}{45}\right), kx \cdot kx, \frac{-1}{3}\right), \color{blue}{kx \cdot kx}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       18. unpow2 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{315}, kx \cdot kx, \frac{2}{45}\right), kx \cdot kx, \frac{-1}{3}\right), kx \cdot kx, 1\right) \cdot \color{blue}{\left(kx \cdot kx\right)} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       19. lower-*.f64 95.8

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0031746031746031746, kx \cdot kx, 0.044444444444444446\right), kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \color{blue}{\left(kx \cdot kx\right)} + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th \]

   12. Applied rewrites 95.8%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0031746031746031746, kx \cdot kx, 0.044444444444444446\right), kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)} + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th \]

   if 1.44999999999999996 < kx

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, 2 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}}{2}} \cdot \sin th \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      5. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \left(\color{blue}{-2} \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \color{blue}{\left(-2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      10. lower-sqrt.f64 55.8

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \left(-2 \cdot kx\right)} \cdot \color{blue}{\sqrt{2}}}{2}} \cdot \sin th \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(-2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 8.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;kx \leq 1.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0031746031746031746, kx \cdot kx, 0.044444444444444446\right), kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{2} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 59.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;kx \leq 1.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right) + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{2} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8.8e-159)
   (* (/ (sin ky) (fabs (sin ky))) (sin th))
   (if (<= kx 1.45)
     (*
      (/
       (sin ky)
       (sqrt
        (+
         (*
          (fma
           (fma 0.044444444444444446 (* kx kx) -0.3333333333333333)
           (* kx kx)
           1.0)
          (* kx kx))
         (- 0.5 (* (cos (* 2.0 ky)) 0.5)))))
      (sin th))
     (*
      (/ (sin ky) (/ (* (sqrt 2.0) (sqrt (- 1.0 (cos (* -2.0 kx))))) 2.0))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8.8e-159) {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	} else if (kx <= 1.45) {
		tmp = (sin(ky) / sqrt(((fma(fma(0.044444444444444446, (kx * kx), -0.3333333333333333), (kx * kx), 1.0) * (kx * kx)) + (0.5 - (cos((2.0 * ky)) * 0.5))))) * sin(th);
	} else {
		tmp = (sin(ky) / ((sqrt(2.0) * sqrt((1.0 - cos((-2.0 * kx))))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8.8e-159)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	elseif (kx <= 1.45)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(fma(fma(0.044444444444444446, Float64(kx * kx), -0.3333333333333333), Float64(kx * kx), 1.0) * Float64(kx * kx)) + Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5))))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 - cos(Float64(-2.0 * kx))))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8.8e-159], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 1.45], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(0.044444444444444446 * N[(kx * kx), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 8.8e-159

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 79.6

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 79.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - {\cos ky}^{2}}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 56.5

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 56.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   if 8.8e-159 < kx < 1.44999999999999996

   1. Initial program 98.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 97.6

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 97.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot {kx}^{2}} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot \color{blue}{\left(kx \cdot kx\right)} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right)} \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{\left(\frac{-1}{3} \cdot {kx}^{2} + 1\right)} \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {kx}^{2}, 1\right)} \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      9. lower-*.f64 95.1

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

   7. Applied rewrites 95.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      5. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      6. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      8. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky + ky\right)}\right)}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 95.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   9. Applied rewrites 95.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   10. Taylor expanded in kx around 0

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right)} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot {kx}^{2}} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(1 + {kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right)\right) \cdot {kx}^{2}} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       3. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left({kx}^{2} \cdot \left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) + 1\right)} \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       4. *-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}\right) \cdot {kx}^{2}} + 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{2}{45} \cdot {kx}^{2} - \frac{1}{3}, {kx}^{2}, 1\right)} \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       6. sub-neg N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{2}{45} \cdot {kx}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       7. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{2}{45} \cdot {kx}^{2} + \color{blue}{\frac{-1}{3}}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{45}, {kx}^{2}, \frac{-1}{3}\right)}, {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       9. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, \color{blue}{kx \cdot kx}, \frac{-1}{3}\right), {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, \color{blue}{kx \cdot kx}, \frac{-1}{3}\right), {kx}^{2}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       11. unpow2 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, kx \cdot kx, \frac{-1}{3}\right), \color{blue}{kx \cdot kx}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, kx \cdot kx, \frac{-1}{3}\right), \color{blue}{kx \cdot kx}, 1\right) \cdot {kx}^{2} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       13. unpow2 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45}, kx \cdot kx, \frac{-1}{3}\right), kx \cdot kx, 1\right) \cdot \color{blue}{\left(kx \cdot kx\right)} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       14. lower-*.f64 95.6

           \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \color{blue}{\left(kx \cdot kx\right)} + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th \]

   12. Applied rewrites 95.6%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right)} + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th \]

   if 1.44999999999999996 < kx

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, 2 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}}{2}} \cdot \sin th \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      5. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \left(\color{blue}{-2} \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \color{blue}{\left(-2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      10. lower-sqrt.f64 55.8

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \left(-2 \cdot kx\right)} \cdot \color{blue}{\sqrt{2}}}{2}} \cdot \sin th \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(-2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 8.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;kx \leq 1.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, kx \cdot kx, -0.3333333333333333\right), kx \cdot kx, 1\right) \cdot \left(kx \cdot kx\right) + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{2} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;kx \leq 1.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(-2 \cdot ky\right), -0.5, 0.5\right) + \left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{2} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8.8e-159)
   (* (/ (sin ky) (fabs (sin ky))) (sin th))
   (if (<= kx 1.45)
     (*
      (/
       (sin ky)
       (sqrt
        (+
         (fma (cos (* -2.0 ky)) -0.5 0.5)
         (* (* (fma -0.3333333333333333 (* kx kx) 1.0) kx) kx))))
      (sin th))
     (*
      (/ (sin ky) (/ (* (sqrt 2.0) (sqrt (- 1.0 (cos (* -2.0 kx))))) 2.0))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8.8e-159) {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	} else if (kx <= 1.45) {
		tmp = (sin(ky) / sqrt((fma(cos((-2.0 * ky)), -0.5, 0.5) + ((fma(-0.3333333333333333, (kx * kx), 1.0) * kx) * kx)))) * sin(th);
	} else {
		tmp = (sin(ky) / ((sqrt(2.0) * sqrt((1.0 - cos((-2.0 * kx))))) / 2.0)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8.8e-159)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	elseif (kx <= 1.45)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(fma(cos(Float64(-2.0 * ky)), -0.5, 0.5) + Float64(Float64(fma(-0.3333333333333333, Float64(kx * kx), 1.0) * kx) * kx)))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 - cos(Float64(-2.0 * kx))))) / 2.0)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8.8e-159], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 1.45], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] + N[(N[(N[(-0.3333333333333333 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 8.8e-159

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 79.6

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 79.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - {\cos ky}^{2}}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 56.5

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 56.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   if 8.8e-159 < kx < 1.44999999999999996

   1. Initial program 98.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 97.6

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 97.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot {kx}^{2}} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot \color{blue}{\left(kx \cdot kx\right)} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right)} \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{\left(\frac{-1}{3} \cdot {kx}^{2} + 1\right)} \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {kx}^{2}, 1\right)} \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      9. lower-*.f64 95.1

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

   7. Applied rewrites 95.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      5. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      6. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      8. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky + ky\right)}\right)}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 95.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   9. Applied rewrites 95.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}}} \cdot \sin th \]

       2. sub-neg N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)\right)\right)}}} \cdot \sin th \]

       3. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)\right) + \frac{1}{2}\right)}}} \cdot \sin th \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\left(\mathsf{neg}\left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}\right)\right) + \frac{1}{2}\right)}} \cdot \sin th \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\color{blue}{\cos \left(2 \cdot ky\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \frac{1}{2}\right)}} \cdot \sin th \]

       6. lift-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\color{blue}{\cos \left(2 \cdot ky\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}\right)}} \cdot \sin th \]

       7. cos-neg-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot ky\right)\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}\right)}} \cdot \sin th \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\cos \left(\mathsf{neg}\left(\color{blue}{2 \cdot ky}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}\right)}} \cdot \sin th \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot ky\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}\right)}} \cdot \sin th \]

       10. metadata-eval N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\cos \left(\color{blue}{-2} \cdot ky\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}\right)}} \cdot \sin th \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\cos \color{blue}{\left(-2 \cdot ky\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}\right)}} \cdot \sin th \]

       12. lift-cos.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\color{blue}{\cos \left(-2 \cdot ky\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \frac{1}{2}\right)}} \cdot \sin th \]

       13. lower-fma.f64 N/A

           \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\mathsf{fma}\left(\cos \left(-2 \cdot ky\right), \mathsf{neg}\left(\frac{1}{2}\right), \frac{1}{2}\right)}}} \cdot \sin th \]

       14. metadata-eval 95.5

           \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \mathsf{fma}\left(\cos \left(-2 \cdot ky\right), \color{blue}{-0.5}, 0.5\right)}} \cdot \sin th \]

   11. Applied rewrites 95.5%

       \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\mathsf{fma}\left(\cos \left(-2 \cdot ky\right), -0.5, 0.5\right)}}} \cdot \sin th \]

   if 1.44999999999999996 < kx

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, 2 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}}{2}} \cdot \sin th \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      5. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \left(\color{blue}{-2} \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \color{blue}{\left(-2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      10. lower-sqrt.f64 55.8

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \left(-2 \cdot kx\right)} \cdot \color{blue}{\sqrt{2}}}{2}} \cdot \sin th \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(-2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 8.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;kx \leq 1.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\cos \left(-2 \cdot ky\right), -0.5, 0.5\right) + \left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{2} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 59.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;kx \leq 1.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right) + kx \cdot kx}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{2} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}{2}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8.8e-159)
   (* (/ (sin ky) (fabs (sin ky))) (sin th))
   (if (<= kx 1.45)
     (*
      (/ (sin ky) (sqrt (+ (- 0.5 (* (cos (* 2.0 ky)) 0.5)) (* kx kx))))
      (sin th))
     (*
      (/ (sin ky) (/ (* (sqrt 2.0) (sqrt (- 1.0 (cos (* -2.0 kx))))) 2.0))
      (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8.8e-159) {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	} else if (kx <= 1.45) {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * ky)) * 0.5)) + (kx * kx)))) * sin(th);
	} else {
		tmp = (sin(ky) / ((sqrt(2.0) * sqrt((1.0 - cos((-2.0 * kx))))) / 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 <= 8.8d-159) then
        tmp = (sin(ky) / abs(sin(ky))) * sin(th)
    else if (kx <= 1.45d0) then
        tmp = (sin(ky) / sqrt(((0.5d0 - (cos((2.0d0 * ky)) * 0.5d0)) + (kx * kx)))) * sin(th)
    else
        tmp = (sin(ky) / ((sqrt(2.0d0) * sqrt((1.0d0 - cos(((-2.0d0) * kx))))) / 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 <= 8.8e-159) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(ky))) * Math.sin(th);
	} else if (kx <= 1.45) {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (Math.cos((2.0 * ky)) * 0.5)) + (kx * kx)))) * Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / ((Math.sqrt(2.0) * Math.sqrt((1.0 - Math.cos((-2.0 * kx))))) / 2.0)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 8.8e-159:
		tmp = (math.sin(ky) / math.fabs(math.sin(ky))) * math.sin(th)
	elif kx <= 1.45:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (math.cos((2.0 * ky)) * 0.5)) + (kx * kx)))) * math.sin(th)
	else:
		tmp = (math.sin(ky) / ((math.sqrt(2.0) * math.sqrt((1.0 - math.cos((-2.0 * kx))))) / 2.0)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8.8e-159)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	elseif (kx <= 1.45)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * ky)) * 0.5)) + Float64(kx * kx)))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / Float64(Float64(sqrt(2.0) * sqrt(Float64(1.0 - cos(Float64(-2.0 * kx))))) / 2.0)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 8.8e-159)
		tmp = (sin(ky) / abs(sin(ky))) * sin(th);
	elseif (kx <= 1.45)
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * ky)) * 0.5)) + (kx * kx)))) * sin(th);
	else
		tmp = (sin(ky) / ((sqrt(2.0) * sqrt((1.0 - cos((-2.0 * kx))))) / 2.0)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8.8e-159], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[kx, 1.45], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 - N[Cos[N[(-2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 8.8e-159

   1. Initial program 91.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 79.6

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 79.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - {\cos ky}^{2}}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 56.5

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 56.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

   if 8.8e-159 < kx < 1.44999999999999996

   1. Initial program 98.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 97.6

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 97.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot {kx}^{2}} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot \color{blue}{\left(kx \cdot kx\right)} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      3. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) \cdot kx\right)} \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{\left(\frac{-1}{3} \cdot {kx}^{2} + 1\right)} \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {kx}^{2}, 1\right)} \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

      9. lower-*.f64 95.1

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, \color{blue}{kx \cdot kx}, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]

   7. Applied rewrites 95.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx} + \left(1 - \cos ky \cdot \cos ky\right)}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      5. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      6. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

      8. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(ky + ky\right)}\right)}} \cdot \sin th \]

      9. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right) \cdot \frac{1}{2}}\right)}} \cdot \sin th \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \color{blue}{\cos \left(ky + ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      13. lower-*.f64 95.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \left(0.5 - \cos \color{blue}{\left(2 \cdot ky\right)} \cdot 0.5\right)}} \cdot \sin th \]

   9. Applied rewrites 95.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right) \cdot kx\right) \cdot kx + \color{blue}{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}}} \cdot \sin th \]

   10. Taylor expanded in kx around 0

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + \left(\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

       2. lower-*.f64 95.4

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th \]

   12. Applied rewrites 95.4%

       \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + \left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right)}} \cdot \sin th \]

   if 1.44999999999999996 < kx

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.6

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(1 - \cos \left(2 \cdot kx\right), 2, 2 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)\right)}}{2}}} \cdot \sin th \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{1 - \cos \left(2 \cdot kx\right)}}}{2}} \cdot \sin th \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{\color{blue}{1 - \cos \left(2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      5. cos-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot kx\right)\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      8. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \left(\color{blue}{-2} \cdot kx\right)} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \color{blue}{\left(-2 \cdot kx\right)}} \cdot \sqrt{2}}{2}} \cdot \sin th \]

      10. lower-sqrt.f64 55.8

          \[\leadsto \frac{\sin ky}{\frac{\sqrt{1 - \cos \left(-2 \cdot kx\right)} \cdot \color{blue}{\sqrt{2}}}{2}} \cdot \sin th \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{\sin ky}{\frac{\color{blue}{\sqrt{1 - \cos \left(-2 \cdot kx\right)} \cdot \sqrt{2}}}{2}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 8.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;kx \leq 1.45:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot ky\right) \cdot 0.5\right) + kx \cdot kx}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sqrt{2} \cdot \sqrt{1 - \cos \left(-2 \cdot kx\right)}}{2}} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 46.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;ky \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + \left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4.2e-151)
   (/ (sin th) (/ (sin kx) ky))
   (if (<= ky 5.8e-5)
     (*
      (/ (sin ky) (sqrt (+ (* ky ky) (- 0.5 (* (cos (* 2.0 kx)) 0.5)))))
      (sin th))
     (* (/ (sin ky) (fabs (sin ky))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.2e-151) {
		tmp = sin(th) / (sin(kx) / ky);
	} else if (ky <= 5.8e-5) {
		tmp = (sin(ky) / sqrt(((ky * ky) + (0.5 - (cos((2.0 * kx)) * 0.5))))) * sin(th);
	} else {
		tmp = (sin(ky) / fabs(sin(ky))) * 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 (ky <= 4.2d-151) then
        tmp = sin(th) / (sin(kx) / ky)
    else if (ky <= 5.8d-5) then
        tmp = (sin(ky) / sqrt(((ky * ky) + (0.5d0 - (cos((2.0d0 * kx)) * 0.5d0))))) * sin(th)
    else
        tmp = (sin(ky) / abs(sin(ky))) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4.2e-151) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else if (ky <= 5.8e-5) {
		tmp = (Math.sin(ky) / Math.sqrt(((ky * ky) + (0.5 - (Math.cos((2.0 * kx)) * 0.5))))) * Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(ky))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 4.2e-151:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	elif ky <= 5.8e-5:
		tmp = (math.sin(ky) / math.sqrt(((ky * ky) + (0.5 - (math.cos((2.0 * kx)) * 0.5))))) * math.sin(th)
	else:
		tmp = (math.sin(ky) / math.fabs(math.sin(ky))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4.2e-151)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	elseif (ky <= 5.8e-5)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(ky * ky) + Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5))))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 4.2e-151)
		tmp = sin(th) / (sin(kx) / ky);
	elseif (ky <= 5.8e-5)
		tmp = (sin(ky) / sqrt(((ky * ky) + (0.5 - (cos((2.0 * kx)) * 0.5))))) * sin(th);
	else
		tmp = (sin(ky) / abs(sin(ky))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4.2e-151], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 5.8e-5], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(ky * ky), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 4.19999999999999981e-151

   1. Initial program 90.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      8. lower-hypot.f64 99.7

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   7. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

      2. lower-sin.f64 32.7

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sin kx}}{ky}} \]

   9. Applied rewrites 32.7%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

   if 4.19999999999999981e-151 < ky < 5.8e-5

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

      2. lower-*.f64 99.3

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot ky}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + ky \cdot ky}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + ky \cdot ky}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin kx} \cdot \sin kx + ky \cdot ky}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx} + ky \cdot ky}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + ky \cdot ky}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)} + ky \cdot ky}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right) + ky \cdot ky}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}\right) + ky \cdot ky}} \cdot \sin th \]

      9. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(kx + kx\right)} \cdot \frac{1}{2}\right) + ky \cdot ky}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(kx + kx\right)} \cdot \frac{1}{2}\right) + ky \cdot ky}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot \frac{1}{2}\right) + ky \cdot ky}} \cdot \sin th \]

      12. lower-*.f64 89.9

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - \cos \color{blue}{\left(2 \cdot kx\right)} \cdot 0.5\right) + ky \cdot ky}} \cdot \sin th \]

   7. Applied rewrites 89.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)} + ky \cdot ky}} \cdot \sin th \]

   if 5.8e-5 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(\mathsf{neg}\left(\sin ky\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right) \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      5. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{neg}\left(\sin ky\right)\right)}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\sin ky}\right)\right)}} \cdot \sin th \]

      7. cos-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \color{blue}{\cos \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)}}} \cdot \sin th \]

      8. 1-sub-sin-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}}} \cdot \sin th \]

      9. sin-+PI/2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \sin \left(ky + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \cdot \sin th \]

      10. sin-+PI/2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky \cdot \cos ky}\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \color{blue}{\cos ky} \cdot \cos ky\right)}} \cdot \sin th \]

      14. lower-cos.f64 98.5

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \left(1 - \cos ky \cdot \color{blue}{\cos ky}\right)}} \cdot \sin th \]

   4. Applied rewrites 98.5%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(1 - \cos ky \cdot \cos ky\right)}}} \cdot \sin th \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1 - {\cos ky}^{2}}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{1 - \color{blue}{\cos ky \cdot \cos ky}}} \cdot \sin th \]

      2. 1-sub-cos N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      4. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]

      5. lower-sin.f64 58.0

         \[\leadsto \frac{\sin ky}{\left|\color{blue}{\sin ky}\right|} \cdot \sin th \]

   7. Applied rewrites 58.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\left|\sin ky\right|}} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 4.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;ky \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{ky \cdot ky + \left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 10.8% accurate, 39.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (* (* th th) -0.16666666666666666) th))

C

double code(double kx, double ky, double th) {
	return ((th * th) * -0.16666666666666666) * 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 = ((th * th) * (-0.16666666666666666d0)) * th
end function

Java

public static double code(double kx, double ky, double th) {
	return ((th * th) * -0.16666666666666666) * th;
}

Python

def code(kx, ky, th):
	return ((th * th) * -0.16666666666666666) * th

Julia

function code(kx, ky, th)
	return Float64(Float64(Float64(th * th) * -0.16666666666666666) * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = ((th * th) * -0.16666666666666666) * th;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]

   3. +-commutative N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}} + {\sin kx}^{2}}} \cdot \sin th \]

   5. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]

   7. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

   8. lower-hypot.f64 99.7

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

4. Applied rewrites 99.7%

   \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

5. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
6. Step-by-step derivation

   1. lower-sin.f64 24.3

      \[\leadsto \color{blue}{\sin th} \]

7. Applied rewrites 24.3%

   \[\leadsto \color{blue}{\sin th} \]

8. Taylor expanded in th around 0

   \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 12.7%

    \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

11. Taylor expanded in th around inf

    \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
12. Step-by-step derivation

13. Applied rewrites 10.6%

    \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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)))